drmf_bof
'''Orthogonal polynomials'''
{{DISPLAYTITLE:Orthogonal polynomials}}
<div id="drmf_head">
<div id="alignleft"> << [[Orthogonal Polynomials|Orthogonal Polynomials]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Orthogonal polynomials]] </div>
<div id="alignright"> [[The gamma and beta function|The gamma and beta function]] >> </div>
</div>

== Orthogonal polynomials ==

<math id="KLS:01.01:01">{\displaystyle
\int_a^bp_m(x)p_n(x)w(x)\,dx=0,\quad m\neq n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:02">{\displaystyle
0<\int_a^bx^{2n}w(x)\,dx<\infty\quad\textrm{for all}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:03">{\displaystyle
0<\int_a^bx^{2n}\,d\alpha(x)<\infty\quad\textrm{for all}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:04">{\displaystyle
\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:05">{\displaystyle
\sum_{x\in X}p_m(x)p_n(x)w_x=0,\quad m\neq n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots,N\}}</math></div><br />
<math id="KLS:01.01:06">{\displaystyle
\sigma_n=\int_a^b\left\{p_n(x)\right\}^2w(x)\,dx
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.01:07">{\displaystyle
\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.01:08">{\displaystyle
\sigma_n=\sum_{x\in X}\left\{p_n(x)\right\}^2w_x
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots,N}</math></div><br />
<math id="KLS:01.01:09">{\displaystyle
p_n(x)=k_nx^n+\,\textrm{lower order terms}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.01:10">{\displaystyle
\,\Kronecker{m}{n}:=\left\{\begin{array}{ll}0, &m\neq n,\\[5mm]
1, & m=n,\end{array}\right.
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:11">{\displaystyle
\int_a^bp_m(x)p_n(x)w(x)\,dx=\sigma_n\,\Kronecker{m}{n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots\}}</math></div><br />
<math id="KLS:01.01:12">{\displaystyle
\sum_{x\in X}p_m(x)p_n(x)w_x=\sigma_n\,\Kronecker{m}{n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle m,n\in\{0,1,2,\ldots,N\}}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Orthogonal Polynomials|Orthogonal Polynomials]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Orthogonal polynomials]] </div>
<div id="alignright"> [[The gamma and beta function|The gamma and beta function]] >> </div>
</div>

drmf_eof
drmf_bof
'''The gamma and beta function'''
{{DISPLAYTITLE:The gamma and beta function}}
<div id="drmf_head">
<div id="alignleft"> << [[Orthogonal polynomials|Orthogonal polynomials]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The gamma and beta function]] </div>
<div id="alignright"> [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] >> </div>
</div>

== The gamma and beta function ==

<math id="KLS:01.02:01">{\displaystyle
\index{Gamma function}
\EulerGamma@{z}:=\int_0^{\infty}t^{z-1}\expe^{-t}\,dt
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \realpart{z}>0}</math></div><br />
<math id="KLS:01.02:02">{\displaystyle
\EulerGamma@{z+1}=z\EulerGamma@{z}\quad\textrm{with}\quad\EulerGamma@{1}=1
}</math><br />
<math id="KLS:01.02:03">{\displaystyle
\EulerGamma@{z}\EulerGamma@{1-z}=\frac{\cpi}{\sin@{\cpi z}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle z\notin\mathbb{Z}}</math></div><br />
<math id="KLS:01.02:04">{\displaystyle
\int_{-\infty}^{\infty}\expe^{-x^2}\,dx=2\int_0^{\infty}\expe^{-x^2}\,dx
=\int_0^{\infty}t^{-1/2}\expe^{-t}\,dt=\EulerGamma@{1/2}=\sqrt{\cpi}
}</math><br />
<math id="KLS:01.02:05">{\displaystyle
\int_{-\infty}^{\infty}\expe^{-\alpha^2x^2-2\beta x}\,dx
=\sqrt{\frac{\cpi}{\alpha^2}}\,\expe^{\beta^2/\alpha^2}
\quad\alpha,\beta\in\mathbb{R}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \alpha\neq 0}</math></div><br />
<math id="KLS:01.02:06">{\displaystyle
\EulerGamma@{z}\EulerGamma@{z+1/2}=2^{1-2z}\sqrt{\cpi}\,\EulerGamma@{2z},\quad
z\in\mathbb{C}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 2z\neq 0,-1,-2,\ldots}</math></div><br />
<math id="KLS:01.02:07">{\displaystyle
\EulerGamma@{z}\sim\sqrt{2\cpi}\,z^{z-1/2}\expe^{-z}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \realpart{z}\rightarrow\infty}</math></div><br />
<math id="KLS:01.02:08">{\displaystyle
\EulerGamma@{x+\iunit y}\sim\sqrt{2\cpi}\,|y|^{x-1/2}\expe^{-|y|\cpi/2}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |y|\rightarrow\infty}</math></div><br />
<math id="KLS:01.02:09">{\displaystyle
\frac{\EulerGamma@{z+a}}{\EulerGamma@{z+b}}\sim z^{a-b},\quad
a,b\in\mathbb{C}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|\rightarrow\infty}</math></div><br />
<math id="KLS:01.02:10">{\displaystyle
\index{Beta function}
\EulerBeta@{x}{y}:=\int_0^1t^{x-1}(1-t)^{y-1}\,dt,\quad\realpart{x}>0
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \realpart{y}>0}</math></div><br />
<math id="KLS:01.02:11">{\displaystyle
\EulerBeta@{x}{y}=\frac{\EulerGamma@{x}\EulerGamma@{y}}{\EulerGamma@{x+y}}
\quad\realpart{x}>0
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \realpart{y}>0}</math></div><br />
<math id="KLS:01.02:12">{\displaystyle
\frac{1}{2\cpi}\int_{-\infty}^{\infty}\frac{dt}{(r+\iunit t)^{\rho}(s-\iunit t)^{\sigma}}
=\frac{(r+s)^{1-\rho-\sigma}\EulerGamma@{\rho+\sigma-1}}{\EulerGamma@{\rho}\EulerGamma@{\sigma}}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Orthogonal polynomials|Orthogonal polynomials]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The gamma and beta function]] </div>
<div id="alignright"> [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] >> </div>
</div>

drmf_eof
drmf_bof
'''The shifted factorial and binomial coefficients'''
{{DISPLAYTITLE:The shifted factorial and binomial coefficients}}
<div id="drmf_head">
<div id="alignleft"> << [[The gamma and beta function|The gamma and beta function]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The shifted factorial and binomial coefficients]] </div>
<div id="alignright"> [[Hypergeometric functions|Hypergeometric functions]] >> </div>
</div>

== The shifted factorial and binomial coefficients ==

<math id="KLS:01.03:01">{\displaystyle
\index{Shifted factorial}
\pochhammer{a}{0}:=1\quad\textrm{and}\quad \pochhammer{a}{k}:=\prod_{n=1}^k(a+n-1),\quad k=1,2,3,\ldots
}</math><br />
<math id="KLS:01.03:02">{\displaystyle
\pochhammer{1}{n}=n!
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.03:03">{\displaystyle
\index{Binomial coefficient}
\binomial{\alpha}{\beta}:=\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}}
}</math><br />
<math id="KLS:01.03:04">{\displaystyle
\binomial{\alpha}{k}:=\frac{\pochhammer{-\alpha}{k}}{k!}(-1)^k
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.03:05">{\displaystyle
\binomial{n}{k}:=\frac{n!}{k!\,(n-k)!},\quad k=0,1,2,\ldots,n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.03:06">{\displaystyle
\binomial{2n}{n}=\frac{\pochhammer{\frac{1}{2}}{n}}{n!}4^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The gamma and beta function|The gamma and beta function]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The shifted factorial and binomial coefficients]] </div>
<div id="alignright"> [[Hypergeometric functions|Hypergeometric functions]] >> </div>
</div>

drmf_eof
drmf_bof
'''Hypergeometric functions'''
{{DISPLAYTITLE:Hypergeometric functions}}
<div id="drmf_head">
<div id="alignleft"> << [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Hypergeometric functions]] </div>
<div id="alignright"> [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] >> </div>
</div>

== Hypergeometric functions ==

<math id="KLS:01.04:01">{\displaystyle
\index{Hypergeometric function}
\HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}:=
\sum\limits_{k=0}^{\infty}\frac{\pochhammer{a_1,\ldots,a_r}{k}}{\pochhammer{b_1,\ldots,b_s}{k}}
\frac{z^k}{k!}
}</math><br />
<math id="KLS:01.04:02">{\displaystyle
\pochhammer{a_1,\ldots,a_r}{k}:=\pochhammer{a_1}{k}\cdots\pochhammer{a_r}{k}
}</math><br />
<math id="KLS:01.04:03">{\displaystyle
\rho=\left\{\begin{array}{ll}
\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm]
\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm]
\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.
}</math><br />
<math id="KLS:01.04:04">{\displaystyle
\HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{z}
=\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{z}
}</math><br />
<math id="KLS:01.04:05">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{\frac{z}{\lambda}}
=\HyperpFq{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{a_rz}
}</math><br />
<math id="KLS:01.04:06">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{\lambda z}=
\HyperpFq{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{\frac{z}{b_s}}
}</math><br />
<math id="KLS:01.04:07">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\HyperpFq{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{z}
=\HyperpFq{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{\frac{a_rz}{b_s}}
}</math><br />
<math id="KLS:01.04:08">{\displaystyle
\Hahn{N}@{x}{\alpha}{\beta}{N}=\sum_{k=0}^N\frac{\pochhammer{N+\alpha+\beta+1}{k}\pochhammer{-x}{k}}{\pochhammer{\alpha+1}{k}k!}
}</math><br />
<math id="KLS:01.04:09">{\displaystyle
\left[f(t)\right]_N:=\sum_{k=0}^N\frac{f^{(k)}(0)}{k!}t^k
}</math><br />
<math id="KLS:01.04:10">{\displaystyle
\expe^t\,\HyperpFq{1}{1}@@{-x}{-N}{-\frac{t}{p}}=
\sum_{k=0}^{\infty}\frac{t^k}{k!}\sum_{m=0}^x\frac{\pochhammer{-x}{m}}{\pochhammer{-N}{m}m!}
\left(-\frac{t}{p}\right)^m
}</math><br />
<math id="KLS:01.04:11">{\displaystyle
\sum_{n=0}^N\frac{\Krawtchouk{n}@{x}{p}{N}}{n!}t^n
}</math><br />
<math id="KLS:01.04:12">{\displaystyle
\expe^z=\HyperpFq{0}{0}@@{-}{-}{z}
}</math><br />
<math id="KLS:01.04:13">{\displaystyle
\cos@@{z}=\HyperpFq{0}{1}@@{-}{\frac{1}{2}}{-\frac{z^2}{4}}
}</math><br />
<math id="KLS:01.04:14">{\displaystyle
\sin@@{z}=z\,\HyperpFq{0}{1}@@{-}{\frac{3}{2}}{-\frac{z^2}{4}}
}</math><br />
<math id="KLS:01.04:15">{\displaystyle
\index{Bessel function}
\BesselJ{\nu}@{z}:=\frac{\left(\frac{1}{2}z\right)^{\nu}}{\EulerGamma@{\nu+1}}\
\HyperpFq{0}{1}@@{-}{\nu+1}{-\frac{z^2}{4}}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The shifted factorial and binomial coefficients|The shifted factorial and binomial coefficients]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Hypergeometric functions]] </div>
<div id="alignright"> [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] >> </div>
</div>

drmf_eof
drmf_bof
'''The binomial theorem and other summation formulas'''
{{DISPLAYTITLE:The binomial theorem and other summation formulas}}
<div id="drmf_head">
<div id="alignleft"> << [[Hypergeometric functions|Hypergeometric functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The binomial theorem and other summation formulas]] </div>
<div id="alignright"> [[Some integrals|Some integrals]] >> </div>
</div>

== The binomial theorem and other summation formulas ==

<math id="KLS:01.05:01">{\displaystyle
\index{Binomial theorem}\index{Summation formula!Binomial theorem}
\HyperpFq{1}{0}@@{a}{-}{z}=\sum_{n=0}^{\infty}\frac{\pochhammer{a}{n}}{n!}z^n=(1-z)^{-a}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|<1}</math></div><br />
<math id="KLS:01.05:02">{\displaystyle
\HyperpFq{1}{0}@@{-n}{-}{z}=\sum_{k=0}^n\frac{\pochhammer{-n}{k}}{k!}z^k
=\sum_{k=0}^n\binomial{n}{k}(-z)^k=(1-z)^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.05:03">{\displaystyle
\index{Gauss summation formula}\index{Summation formula!Gauss}
\HyperpFq{2}{1}@@{a,b}{c}{1}=
\frac{\EulerGamma@{c}\EulerGamma@{c-a-b}}{\EulerGamma@{c-a}\EulerGamma@{c-b}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \realpart{c-a-b}>0}</math></div><br />
<math id="KLS:01.05:04">{\displaystyle
\index{Vandermonde summation formula}\index{Summation formula!Vandermonde}\index{Chu-Vandermonde summation formula}\index{Summation formula!Chu-Vandermonde}
\HyperpFq{2}{1}@@{-n,b}{c}{1}=\frac{\pochhammer{c-b}{n}}{\pochhammer{c}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.05:05">{\displaystyle
\index{Saalsch\"{u}tz summation formula}\index{Summation formula!Saalsch\"{u}tz}\index{Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!Pfaff-Saalsch\"{u}tz}
\HyperpFq{3}{2}@@{-n,a,b}{c,1+a+b-c-n}{1}=\frac{\pochhammer{c-a}{n}\pochhammer{c-b}{n}}{\pochhammer{c}{n}\pochhammer{c-a-b}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.05:06">{\displaystyle
\index{Summation formula!for a very-well-poised $\HyperpFq{5}{4}$}
\HyperpFq{5}{4}@@{1+a/2,a,b,c,d}{a/2,1+a-b,1+a-c,1+a-d}{1}
{}=\frac{\EulerGamma@{1+a-b}\EulerGamma@{1+a-c}\EulerGamma@{1+a-d}\EulerGamma@{1+a-b-c-d}}
{\EulerGamma@{1+a}\EulerGamma@{1+a-b-c}\EulerGamma@{1+a-b-d}\EulerGamma@{1+a-c-d}}
}</math><br />
<math id="KLS:01.05:07">{\displaystyle
\HyperpFq{4}{3}@@{1+a/2,a,b,c}{a/2,1+a-b,1+a-c}{-1}
=\frac{\EulerGamma@{1+a-b}\EulerGamma@{1+a-c}}{\EulerGamma@{1+a}\EulerGamma@{1+a-b-c}}
}</math><br />
<math id="KLS:01.05:08">{\displaystyle
\index{Dougall's bilateral sum}\index{Summation formula!Dougall}
\sum_{n=-\infty}^{\infty}\frac{\EulerGamma@{n+a}\EulerGamma@{n+b}}{\EulerGamma@{n+c}\EulerGamma@{n+d}}
=\frac{\EulerGamma@{a}\EulerGamma@{1-a}\EulerGamma@{b}\EulerGamma@{1-b}\EulerGamma@{c+d-a-b-1}}
{\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}}
}</math><br />
<math id="KLS:01.05:09">{\displaystyle
\EulerGamma@{n+a}\EulerGamma@{n+b}=\frac{\EulerGamma@{a}\EulerGamma@{1-a}\EulerGamma@{b}\EulerGamma@{1-b}}
{\EulerGamma@{1-a-n}\EulerGamma@{1-b-n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n\in\mathbb{Z}}</math></div><br />
<math id="KLS:01.05:10">{\displaystyle
\sum_{n=-\infty}^{\infty}\frac{1}{\EulerGamma@{n+c}\EulerGamma@{n+d}\EulerGamma@{1-a-n}\EulerGamma@{1-b-n}}
{}=\frac{\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c-a}\EulerGamma@{d-a}\EulerGamma@{c-b}\EulerGamma@{d-b}}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Hypergeometric functions|Hypergeometric functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The binomial theorem and other summation formulas]] </div>
<div id="alignright"> [[Some integrals|Some integrals]] >> </div>
</div>

drmf_eof
drmf_bof
'''Some integrals'''
{{DISPLAYTITLE:Some integrals}}
<div id="drmf_head">
<div id="alignleft"> << [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Some integrals]] </div>
<div id="alignright"> [[Transformation formulas|Transformation formulas]] >> </div>
</div>

== Some integrals ==

<math id="KLS:01.06:01">{\displaystyle
\index{Euler's integral representation for a $\HyperpFq{2}{1}$}\index{Hypergeometric function!Euler's integral representation}
\HyperpFq{2}{1}@@{a,b}{c}{z}=\frac{\EulerGamma@{c}}{\EulerGamma@{b}\EulerGamma@{c-b}}
\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}\,dt
}</math><br />
<math id="KLS:01.06:02">{\displaystyle
\index{Barnes' integral representation}\index{Hypergeometric function!Barnes' integral representation}
\frac{\EulerGamma@{a}\EulerGamma@{b}}{\EulerGamma@{c}}\,\HyperpFq{2}{1}@@{a,b}{c}{z}=\frac{1}{2\cpi \iunit}
\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{-s}}{\EulerGamma@{c+s}}(-z)^s\,ds
}</math><br />
<math id="KLS:01.06:03">{\displaystyle
\index{Mellin-Barnes integral}\index{Barnes' first lemma}
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c-s}\EulerGamma@{d-s}\,ds
{}=\frac{\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}}{\EulerGamma@{a+b+c+d}}
}</math><br />
<math id="KLS:01.06:04">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}
\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{1-d-s}\EulerGamma@{-s}}{\EulerGamma@{e+s}}\,ds
{}=\frac{\EulerGamma@{a}\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{1+a-d}\EulerGamma@{1+b-d}\EulerGamma@{1+c-d}}
{\EulerGamma@{e-a}\EulerGamma@{e-b}\EulerGamma@{e-c}}
}</math><br />
<math id="KLS:01.06:05">{\displaystyle
\index{Bailey's integral}
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}
\frac{\EulerGamma@{1+a/2+s}\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}\EulerGamma@{b-a-s}\EulerGamma@{-s}}
{\EulerGamma@{a/2+s}\EulerGamma@{1+a-c+s}\EulerGamma@{1+a-d+s}}\,ds
{}=\frac{\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{d}\EulerGamma@{b+c-a}\EulerGamma@{b+d-a}}
{2\,\EulerGamma@{1+a-c-d}\EulerGamma@{b+c+d-a}}
}</math><br />
<math id="KLS:01.06:06">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}
\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}}{\EulerGamma@{a+2s}}
{}\frac{\EulerGamma@{-s}\EulerGamma@{b-a-s}\EulerGamma@{c-a-s}\EulerGamma@{d-a-s}}{\EulerGamma@{-a-2s}}\,ds
{}=\frac{2\,\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{d}\EulerGamma@{b+c-a}\EulerGamma@{b+d-a}\EulerGamma@{c+d-a}}{\EulerGamma@{b+c+d-a}}
}</math><br />
<math id="KLS:01.06:07">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}
\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{d+s}\EulerGamma@{a-s}\EulerGamma@{b-s}\EulerGamma@{c-s}\EulerGamma@{d-s}}
{\EulerGamma@{2s}\EulerGamma@{-2s}}\,ds
{}=\frac{2\,\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}\EulerGamma@{c+d}}{\EulerGamma@{a+b+c+d}}
}</math><br />
<math id="KLS:01.06:08">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\EulerGamma@{a+s}\EulerGamma@{b-s}z^{-s}\,ds
=\EulerGamma@{a+b}\frac{z^a}{(1+z)^{a+b}}
}</math><br />
<math id="KLS:01.06:09">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}\frac{\EulerGamma@{a+s}\EulerGamma@{b-s}}{\EulerGamma@{c+s}\EulerGamma@{d-s}}\,ds
=\frac{\EulerGamma@{a+b}\EulerGamma@{c+d-a-b-1}}{\EulerGamma@{c+d-1}\EulerGamma@{c-a}\EulerGamma@{d-b}}
}</math><br />
<math id="KLS:01.06:10">{\displaystyle
\frac{1}{2\cpi \iunit}\int_{-\iunit\infty}^{\iunit\infty}
\frac{\EulerGamma@{a+s}\EulerGamma@{b+s}\EulerGamma@{c+s}\EulerGamma@{-s}\EulerGamma@{b-a-s}\EulerGamma@{c-a-s}}
{\EulerGamma@{a+2s}\EulerGamma@{-a-2s}}\,ds
{}=\frac{1}{2}\EulerGamma@{b}\EulerGamma@{c}\EulerGamma@{b+c-a}
}</math><br />
<math id="KLS:01.06:11">{\displaystyle
\index{Wilson's integral}
\frac{1}{2\cpi}\int_0^{\infty}
\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}\EulerGamma@{d+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2\,dx
{}=\frac{\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{a+d}\EulerGamma@{b+c}\EulerGamma@{b+d}\EulerGamma@{c+d}}{\EulerGamma@{a+b+c+d}}
}</math><br />
<math id="KLS:01.06:12">{\displaystyle
\frac{1}{2\cpi}\int_0^{\infty}
\left|\frac{\EulerGamma@{a+\iunit x}\EulerGamma@{b+\iunit x}\EulerGamma@{c+\iunit x}}{\EulerGamma@{2\iunit x}}\right|^2\,dx
=\EulerGamma@{a+b}\EulerGamma@{a+c}\EulerGamma@{b+c}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The binomial theorem and other summation formulas|The binomial theorem and other summation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Some integrals]] </div>
<div id="alignright"> [[Transformation formulas|Transformation formulas]] >> </div>
</div>

drmf_eof
drmf_bof
'''Transformation formulas'''
{{DISPLAYTITLE:Transformation formulas}}
<div id="drmf_head">
<div id="alignleft"> << [[Some integrals|Some integrals]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Transformation formulas]] </div>
<div id="alignright"> [[The q-shifted factorial|The q-shifted factorial]] >> </div>
</div>

== Transformation formulas ==

<math id="KLS:01.07:01">{\displaystyle
\index{Euler's transformation formula}\index{Transformation formula!Euler}
\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z}
}</math><br />
<math id="KLS:01.07:02">{\displaystyle
\index{Pfaff-Kummer transformation formula}\index{Transformation formula!Pfaff-Kummer}
\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{-a}\,\HyperpFq{2}{1}@@{a,c-b}{c}{\frac{z}{z-1}}
}</math><br />
<math id="KLS:01.07:03">{\displaystyle
\index{Kummer's transformation formula}\index{Transformation formula!Kummer}
\HyperpFq{1}{1}@@{a}{c}{z}=\expe^z\,\HyperpFq{1}{1}@@{c-a}{c}{-z}
}</math><br />
<math id="KLS:01.07:04">{\displaystyle
\HyperpFq{1}{1}@@{-n}{a}{x}=\frac{(-x)^n}{\pochhammer{a}{n}}\,\HyperpFq{2}{0}@@{-n,-a-n+1}{-}{-\frac{1}{x}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.07:05">{\displaystyle
\HyperpFq{2}{1}@@{-n,b}{c}{x}=\frac{\pochhammer{b}{n}}{\pochhammer{c}{n}}(-x)^n\,\HyperpFq{2}{1}@@{-n,-c-n+1}{-b-n+1}{\frac{1}{x}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.07:06">{\displaystyle
\index{Whipple's transformation formula}\index{Transformation formula!Whipple}
\HyperpFq{4}{3}@@{-n,a,b,c}{d,e,f}{1}
{}=\frac{\pochhammer{e-a}{n}\pochhammer{f-a}{n}}{\pochhammer{e}{n}\pochhammer{f}{n}}\HyperpFq{4}{3}@@{-n,a,d-b,d-c}{d,a-e-n+1,a-f-n+1}{1}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Some integrals|Some integrals]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Transformation formulas]] </div>
<div id="alignright"> [[The q-shifted factorial|The q-shifted factorial]] >> </div>
</div>

drmf_eof
drmf_bof
'''The q-shifted factorial'''
{{DISPLAYTITLE:The ''q''-shifted factorial}}
<div id="drmf_head">
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-shifted factorial]] </div>
<div id="alignright"> [[The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] >> </div>
</div>

== The ''q''-shifted factorial ==

<math id="KLS:01.08:01">{\displaystyle
\lim\limits_{q\rightarrow 1}\frac{1-q^{\alpha}}{1-q}=\alpha
}</math><br />
<math id="KLS:01.08:02">{\displaystyle
[\alpha]:=\frac{1-q^{\alpha}}{1-q}
}</math><br />
<math id="KLS:01.08:03">{\displaystyle
[0]=0,\quad [n]=\frac{1-q^n}{1-q}=\sum_{k=0}^{n-1}q^k
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=1,2,3,\ldots}</math></div><br />
<math id="KLS:01.08:04">{\displaystyle
\index{q-Shifted factorial@$q$-Shifted factorial}
\qPochhammer{a}{q}{0}:=1\quad\textrm{and}\quad \qPochhammer{a}{q}{k}:=\prod_{n=1}^k(1-aq^{n-1}),\quad k=1,2,3,\ldots
}</math><br />
<math id="KLS:01.08:05">{\displaystyle
\lim\limits_{q\rightarrow 1}\frac{\qPochhammer{q^{\alpha}}{q}{k}}{(1-q)^k}=\pochhammer{\alpha}{k}
}</math><br />
<math id="KLS:01.08:06">{\displaystyle
\qPochhammer{a}{q}{-k}:=\frac{1}{\displaystyle\prod_{n=1}^{k}(1-aq^{-n})},\quad a\neq q,q^2,q^3,\ldots,q^k
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=1,2,3,\ldots}</math></div><br />
<math id="KLS:01.08:07">{\displaystyle
\qPochhammer{a}{q}{-n}=\frac{1}{\qPochhammer{aq^{-n}}{q}{n}}=\frac{(-qa^{-1})^n}{\qPochhammer{qa^{-1}}{q}{n}}
q^{\binomial{n}{2}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math> &<br />
<math>{\displaystyle a\neq 0}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{aq^{-n}}{q}{n}=\qPochhammer{a^{-1}q}{q}{n}(-a)^nq^{-n-\binomial{n}{2}}}</math></div><br />
<math id="KLS:01.08:08">{\displaystyle
\qPochhammer{a}{q^{-1}}{n}=\qPochhammer{a^{-1}}{q}{n}(-a)^nq^{-\binomial{n}{2}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math></div><br />
<math id="KLS:01.08:11">{\displaystyle
\qPochhammer{a}{q}{\lambda}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^{\lambda}}{q}{\infty}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:12">{\displaystyle
\qPochhammer{a}{q}{n+k}=\qPochhammer{a}{q}{n}\qPochhammer{aq^n}{q}{k}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^n}{q}{\infty}}
=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^nq^{\binomial{n}{2}}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:13">{\displaystyle
\frac{\qPochhammer{aq^n}{q}{k}}{\qPochhammer{aq^k}{q}{n}}=\frac{\qPochhammer{a}{q}{k}}{\qPochhammer{a}{q}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^n}{q}{\infty}}
=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^nq^{\binomial{n}{2}}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:14">{\displaystyle
\qPochhammer{aq^k}{q}{n-k}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{a}{q}{k}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots,n}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math> &<br />
<math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^n}{q}{\infty}}
=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^nq^{\binomial{n}{2}}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:17">{\displaystyle
\frac{\qPochhammer{aq^{-n}}{q}{n}}{\qPochhammer{bq^{-n}}{q}{n}}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{b^{-1}q}{q}{n}}
\left(\frac{a}{b}\right)^n,\quad a\neq 0
}</math>
<div align="right">Constraint(s): <math>{\displaystyle b\neq 0}</math> &<br />
<math>{\displaystyle a\neq 0}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{aq^{-n}}{q}{n}=\qPochhammer{a^{-1}q}{q}{n}(-a)^nq^{-n-\binomial{n}{2}}}</math></div><br />
<math id="KLS:01.08:19">{\displaystyle
\frac{\qPochhammer{a}{q}{n-k}}{\qPochhammer{b}{q}{n-k}}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{b}{q}{n}}
\frac{\qPochhammer{b^{-1}q^{1-n}}{q}{k}}{\qPochhammer{a^{-1}q^{1-n}}{q}{k}}\left(\frac{b}{a}\right)^k
{}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle b\neq 0}</math> &<br />
<math>{\displaystyle k=0,1,2,\ldots,n}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math> &<br />
<math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{n-k}=\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{a^{-1}q^{1-n}}{q}{k}}\left(-\frac{q}{a}\right)^k
q^{\binomial{k}{2}-nk},\quad a\neq 0}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^n}{q}{\infty}}
=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^nq^{\binomial{n}{2}}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:20">{\displaystyle
\qPochhammer{q^{-n}}{q}{k}=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{n-k}}(-1)^kq^{\binomial{k}{2}-nk}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots,n}</math></div><br />
<math id="KLS:01.08:21">{\displaystyle
\qPochhammer{aq^{-n}}{q}{k}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{a^{-1}q^{1-k}}{q}{n}}\qPochhammer{a}{q}{k}q^{-nk}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math></div><br />
<math id="KLS:01.08:22">{\displaystyle
\qPochhammer{aq^{-n}}{q}{n-k}=\frac{\qPochhammer{a^{-1}q}{q}{n}}{\qPochhammer{a^{-1}q}{q}{k}}
\left(-\frac{a}{q}\right)^{n-k}q^{\binomial{k}{2}-\binomial{n}{2}}
{}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle k=0,1,2,\ldots,n}</math></div><br />
<math id="KLS:01.08:23">{\displaystyle
\qPochhammer{a}{q}{2n}=\qPochhammer{a}{q^2}{n}\qPochhammer{aq}{q^2}{n}
}</math><br />
<math id="KLS:01.08:26">{\displaystyle
\qPochhammer{a^2}{q^2}{\infty}=\qPochhammer{a}{q}{\infty}\qPochhammer{-a}{q}{\infty}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<math id="KLS:01.08:27">{\displaystyle
\frac{1-a^2q^{2n}}{1-a^2}=\frac{\qPochhammer{a^2q^2}{q^2}{n}}{\qPochhammer{a^2}{q^2}{n}}
=\frac{\qPochhammer{aq}{q}{n}\qPochhammer{-aq}{q}{n}}{\qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle a\neq 0}</math> &<br />
<math>{\displaystyle 0<|q|<1}</math></div><br />
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a^2}{q^2}{n}=\qPochhammer{a}{q}{n}\qPochhammer{-a}{q}{n}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{n}=\frac{\qPochhammer{a}{q}{\infty}}{\qPochhammer{aq^n}{q}{\infty}}
=\qPochhammer{a^{-1}q^{1-n}}{q}{n}(-a)^nq^{\binomial{n}{2}}}</math> &<br />
<math>{\displaystyle \qPochhammer{a}{q}{\infty}=\prod_{k=0}^{\infty}(1-aq^k)
=\qPochhammer{a}{q^2}{\infty}\qPochhammer{aq}{q^2}{\infty}}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-shifted factorial]] </div>
<div id="alignright"> [[The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] >> </div>
</div>

drmf_eof
drmf_bof
'''The q-gamma function and q-binomial coefficients'''
{{DISPLAYTITLE:The ''q''-gamma function and ''q''-binomial coefficients}}
<div id="drmf_head">
<div id="alignleft"> << [[The q-shifted factorial|The q-shifted factorial]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-gamma function and q-binomial coefficients]] </div>
<div id="alignright"> [[Basic hypergeometric functions|Basic hypergeometric functions]] >> </div>
</div>

== The ''q''-gamma function and ''q''-binomial coefficients ==

<math id="KLS:01.09:01">{\displaystyle
\index{q-Gamma function@$q$-Gamma function}
\qGamma{q}@{x}:=\frac{\qPochhammer{q}{q}{\infty}}{\qPochhammer{q^x}{q}{\infty}}(1-q)^{1-x}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.09:02">{\displaystyle
\lim\limits_{q\rightarrow 1}\qGamma{q}@{x}=\EulerGamma@{x}
}</math><br />
<math id="KLS:01.09:03">{\displaystyle
\qGamma{q}@{z+1}=\frac{1-q^z}{1-q}\qGamma{q}@{z}\quad\textrm{with}\quad\qGamma{q}@{1}=1
}</math><br />
<math id="KLS:01.09:04">{\displaystyle
\qGamma{q}@{x}=\frac{\qPochhammer{q^{-1}}{q^{-1}}{\infty}}{\qPochhammer{q^{-x}}{q^{-1}}{\infty}}
q^{\binomial{x}{2}}(q-1)^{1-x}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle q>1}</math></div><br />
<math id="KLS:01.09:05">{\displaystyle
\index{q-Binomial coefficient@$q$-Binomial coefficient}
\qBinomial{n}{k}{q}:=\frac{\qPochhammer{q}{q}{n}}{\qPochhammer{q}{q}{k}\qPochhammer{q}{q}{n-k}}=\qBinomial{n}{n-k}{q}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle k=0,1,2,\ldots,n}</math></div><br />
<math id="KLS:01.09:06">{\displaystyle
\qBinomial{\alpha}{k}{q}:=\frac{\qPochhammer{q^{-\alpha}}{q}{k}}{\qPochhammer{q}{q}{k}}
(-1)^kq^{k\alpha-\binomial{k}{2}}
}</math><br />
<math id="KLS:01.09:07">{\displaystyle
\qBinomial{\alpha}{\beta}{q}:=\frac{\qGamma{q}@{\alpha+1}}{\qGamma{q}@{\beta+1}\qGamma{q}@{\alpha-\beta+1}}
=\frac{\qPochhammer{q^{\beta+1}}{q}{\infty}\qPochhammer{q^{\alpha-\beta+1}}{q}{\infty}}
{\qPochhammer{q}{q}{\infty}\qPochhammer{q^{\alpha+1}}{q}{\infty}}
}</math><br />
<math id="KLS:01.09:08">{\displaystyle
\frac{\qPochhammer{q^{\alpha+1}}{q}{n}}{\qPochhammer{q}{q}{n}}=\qBinomial{n+\alpha}{n}{q}
}</math><br />
<math id="KLS:01.09:09">{\displaystyle
\lim\limits_{q\rightarrow 1}\qBinomial{\alpha}{\beta}{q}
=\frac{\EulerGamma@{\alpha+1}}{\EulerGamma@{\beta+1}\EulerGamma@{\alpha-\beta+1}}=\binomial{\alpha}{\beta}
}</math><br />
<math id="KLS:01.09:10">{\displaystyle
\frac{1}{\qPochhammer{q}{q}{n}}=\sum_{k=0}^n\frac{q^k}{\qPochhammer{q}{q}{k}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.09:11">{\displaystyle
\qPochhammer{a}{q}{n}=\sum_{k=0}^n\qBinomial{n}{k}{q}q^{\binomial{k}{2}}(-a)^k
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The q-shifted factorial|The q-shifted factorial]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-gamma function and q-binomial coefficients]] </div>
<div id="alignright"> [[Basic hypergeometric functions|Basic hypergeometric functions]] >> </div>
</div>

drmf_eof
drmf_bof
'''Basic hypergeometric functions'''
{{DISPLAYTITLE:Basic hypergeometric functions}}
<div id="drmf_head">
<div id="alignleft"> << [[The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Basic hypergeometric functions]] </div>
<div id="alignright"> [[The q-binomial theorem and other summation formulas|The q-binomial theorem and other summation formulas]] >> </div>
</div>

== Basic hypergeometric functions ==

<math id="KLS:01.10:01">{\displaystyle
\index{Basic hypergeometric function}
\qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{z}
{}:=\sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_r}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}}
(-1)^{(1+s-r)k}q^{(1+s-r)\binomial{k}{2}}\frac{z^k}{\qPochhammer{q}{q}{k}}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a_1,\ldots,a_r}{q}{k}:=\qPochhammer{a_1}{q}{k}\cdots\qPochhammer{a_r}{q}{k}}</math></div><br />
<math id="KLS:01.10:03">{\displaystyle
\rho=\left\{\begin{array}{ll}
\displaystyle \infty & \quad\textrm{if}\quad r < s+1\\[5mm]
\displaystyle 1 & \quad\textrm{if}\quad r = s+1\\[5mm]
\displaystyle 0 & \quad\textrm{if}\quad r > s+1.\end{array}\right.
}</math><br />
<math id="KLS:01.10:04">{\displaystyle
\qHyperrphis{s+1}{s}@@{a_1,\ldots,a_{s+1}}{b_1,\ldots,b_s}{q}{z}=
\sum\limits_{k=0}^{\infty}\frac{\qPochhammer{a_1,\ldots,a_{s+1}}{q}{k}}{\qPochhammer{b_1,\ldots,b_s}{q}{k}}
\frac{z^k}{\qPochhammer{q}{q}{k}}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle \qPochhammer{a_1,\ldots,a_r}{q}{k}:=\qPochhammer{a_1}{q}{k}\cdots\qPochhammer{a_r}{q}{k}}</math></div><br />
<math id="KLS:01.10:05">{\displaystyle
\lim\limits_{q\rightarrow 1}
\qHyperrphis{r}{s}@@{q^{a_1},\ldots,q^{a_r}}{q^{b_1},\ldots,q^{b_s}}{q}{(q-1)^{1+s-r}z}
=\HyperpFq{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{z}
}</math><br />
<math id="KLS:01.10:06">{\displaystyle
\lim\limits_{a_r\rightarrow\infty}
\qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_s}{q}{\frac{z}{a_r}}=
\qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{z}
}</math><br />
<math id="KLS:01.10:07">{\displaystyle
\qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\mu}{b_1,\ldots,b_{s-1},\mu}{q}{z}=
\qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{z}
}</math><br />
<math id="KLS:01.10:08">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_s}{q}{\frac{z}{\lambda}}
=\qHyperrphis{r-1}{s}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_s}{q}{a_rz}
}</math><br />
<math id="KLS:01.10:09">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\qHyperrphis{r}{s}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{\lambda z}=
\qHyperrphis{r}{s-1}@@{a_1,\ldots,a_r}{b_1,\ldots,b_{s-1}}{q}{\frac{z}{b_s}}
}</math><br />
<math id="KLS:01.10:10">{\displaystyle
\lim\limits_{\lambda\rightarrow\infty}
\qHyperrphis{r}{s}@@{a_1,\ldots,a_{r-1},\lambda a_r}{b_1,\ldots,b_{s-1},\lambda b_s}{q}{z}
=\qHyperrphis{r-1}{s-1}@@{a_1,\ldots,a_{r-1}}{b_1,\ldots,b_{s-1}}{q}{\frac{a_rz}{b_s}}
}</math><br />
<math id="KLS:01.10:11">{\displaystyle
\qHahn{N}@{q^{-x}}{\alpha}{\beta}{N}{q}=\sum_{k=0}^N
\frac{\qPochhammer{\alpha\beta q^{N+1}}{q}{k}\qPochhammer{q^{-x}}{q}{k}}{\qPochhammer{\alpha q}{q}{k}\qPochhammer{q}{q}{k}}q^k
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The q-gamma function and q-binomial coefficients|The q-gamma function and q-binomial coefficients]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Basic hypergeometric functions]] </div>
<div id="alignright"> [[The q-binomial theorem and other summation formulas|The q-binomial theorem and other summation formulas]] >> </div>
</div>

drmf_eof
drmf_bof
'''The q-binomial theorem and other summation formulas'''
{{DISPLAYTITLE:The ''q''-binomial theorem and other summation formulas}}
<div id="drmf_head">
<div id="alignleft"> << [[Basic hypergeometric functions|Basic hypergeometric functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-binomial theorem and other summation formulas]] </div>
<div id="alignright"> [[More integrals|More integrals]] >> </div>
</div>

== The ''q''-binomial theorem and other summation formulas ==

<math id="KLS:01.11:01">{\displaystyle
\index{q-Binomial theorem@$q$-Binomial theorem}\index{Summation formula!q-Binomial theorem@$q$-Binomial theorem}
\qHyperrphis{1}{0}@@{a}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{\qPochhammer{a}{q}{n}}{\qPochhammer{q}{q}{n}}z^n=
\frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|<1}</math></div><br />
<math id="KLS:01.11:02">{\displaystyle
\qHyperrphis{1}{0}@@{q^{-n}}{-}{q}{z}=\qPochhammer{zq^{-n}}{q}{n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:03">{\displaystyle
\index{q-Gauss summation formula@$q$-Gauss summation formula}\index{Summation formula!q-Gauss@$q$-Gauss}
\qHyperrphis{2}{1}@@{a,b}{c}{q}{\frac{c}{ab}}=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{\infty}}
{\qPochhammer{c,a^{-1}b^{-1}c}{q}{\infty}},\quad 0<|q|<1
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \left|\frac{c}{ab}\right|<1}</math></div><br />
<math id="KLS:01.11:04">{\displaystyle
\index{q-Vandermonde summation formula@$q$-Vandermonde summation formula}\index{Summation formula!q-Vandermonde@$q$-Vandermonde}\index{q-Chu-Vandermonde summation formula@$q$-Chu-Vandermonde summation formula}\index{Summation formula!q-Chu-Vandermonde@$q$-Chu-Vandermonde}
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{\frac{cq^n}{b}}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:05">{\displaystyle
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{q}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}b^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:06">{\displaystyle
\index{Summation formula!for a $\qHyperrphis{1}{1}$}
\qHyperrphis{1}{1}@@{a}{c}{q}{\frac{c}{a}}=\frac{\qPochhammer{a^{-1}c}{q}{\infty}}{\qPochhammer{c}{q}{\infty}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br />
<math id="KLS:01.11:07">{\displaystyle
\index{Summation formula!for a terminating $\qHyperrphis{2}{0}$}
\qHyperrphis{2}{0}@@{q^{-n},b}{-}{q}{\frac{q^n}{b}}=\frac{1}{b^n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:08">{\displaystyle
\index{Summation formula!for a $\qHyperrphis{0}{1}$}
\qHyperrphis{0}{1}@@{-}{c}{q}{c}=\frac{1}{\qPochhammer{c}{q}{\infty}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br />
<math id="KLS:01.11:09">{\displaystyle
\index{q-Saalschutz summation formula@$q$-Saalsch\"{u}tz summation formula}\index{Summation formula!q-Saalschutz@$q$-Saalsch\"{u}tz}\index{q-Pfaff-Saalschutz summation formula@$q$-Pfaff-Saalsch\"{u}tz summation formula}\index{Summation formula!q-Pfaff-Saalschutz@$q$-Pfaff-Saalsch\"{u}tz}
\qHyperrphis{3}{2}@@{q^{-n},a,b}{c,abc^{-1}q^{1-n}}{q}{q}=\frac{\qPochhammer{a^{-1}c,b^{-1}c}{q}{n}}
{\qPochhammer{c,a^{-1}b^{-1}c}{q}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:10">{\displaystyle
\index{Jackson's summation formula}\index{Summation formula!Jackson}\index{Summation formula!for a very-well-poised $\qHyperrphis{6}{5}$}
\qHyperrphis{6}{5}@@{q\sqrt{a},-q\sqrt{a},a,b,c,d}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,ad^{-1}q}{q}{\frac{aq}{bcd}}
{}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q,ab^{-1}d^{-1}q,ac^{-1}d^{-1}q}{q}{\infty}}
{\qPochhammer{ab^{-1}q,ac^{-1}q,ad^{-1}q,ab^{-1}c^{-1}d^{-1}q}{q}{\infty}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle \left|\frac{aq}{bcd}\right|<1}</math></div><br />
<math id="KLS:01.11:11">{\displaystyle
\qHyperrphis{6}{5}@@{q\sqrt{a},-q\sqrt{a},a,b,c,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,ac^{-1}q,aq^{n+1}}{q}{\frac{aq^{n+1}}{bc}}
{}=\frac{\qPochhammer{aq,ab^{-1}c^{-1}q}{q}{n}}{\qPochhammer{ab^{-1}q,ac^{-1}q}{q}{n}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:12">{\displaystyle
\qHyperrphis{6}{4}@@{q\sqrt{a},-q\sqrt{a},a,b,0,q^{-n}}{\sqrt{a},-\sqrt{a},ab^{-1}q,aq^{n+1}}{q}{\frac{q^n}{b}}
{}=\frac{\qPochhammer{aq}{q}{n}}{\qPochhammer{ab^{-1}q}{q}{n}}b^{-n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.11:13">{\displaystyle
\qHyperrphis{6}{3}@@{q\sqrt{a},-q\sqrt{a},a,0,0,q^{-n}}{\sqrt{a},-\sqrt{a},aq^{n+1}}{q}{\frac{q^{n-1}}{a}}
{}=(-1)^na^{-n}q^{-\binomial{n+1}{2}}\qPochhammer{aq}{q}{n}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Basic hypergeometric functions|Basic hypergeometric functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-binomial theorem and other summation formulas]] </div>
<div id="alignright"> [[More integrals|More integrals]] >> </div>
</div>

drmf_eof
drmf_bof
'''More integrals'''
{{DISPLAYTITLE:More integrals}}
<div id="drmf_head">
<div id="alignleft"> << [[The q-binomial theorem and other summation formulas|The q-binomial theorem and other summation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|More integrals]] </div>
<div id="alignright"> [[Transformation formulas|Transformation formulas]] >> </div>
</div>

== More integrals ==

<math id="KLS:01.12:01">{\displaystyle
\int_0^{\infty}x^{c-1}\frac{\qPochhammer{-ax,-bq/x}{q}{\infty}}{\qPochhammer{-x,-q/x}{q}{\infty}}\,dx
=\frac{\cpi}{\sin@{\cpi c}}\,\frac{\qPochhammer{ab,q^c,q^{1-c}}{q}{\infty}}{\qPochhammer{bq^c,aq^{-c},q}{q}{\infty}}
}</math><br />
<math id="KLS:01.12:02">{\displaystyle
\int_0^{\infty}x^{c-1}\frac{\qPochhammer{-ax}{q}{\infty}}{\qPochhammer{-x}{q}{\infty}}\,dx
=\frac{\cpi}{\sin@{\cpi c}}\,\frac{\qPochhammer{a,q^{1-c}}{q}{\infty}}{\qPochhammer{aq^{-c},q}{q}{\infty}}
}</math><br />
<math id="KLS:01.12:03">{\displaystyle
\int_0^{\infty}\frac{\qPochhammer{-ax,-bq/x}{q}{\infty}}{\qPochhammer{-x,-q/x}{q}{\infty}}\,dx
=-\ln@@{q}\,\frac{\qPochhammer{ab,q}{q}{\infty}}{\qPochhammer{bq,a/q}{q}{\infty}}
}</math><br />
<math id="KLS:01.12:04">{\displaystyle
\index{Askey-Wilson integral}\index{Askey-Wilson q-beta integral@Askey-Wilson $q$-beta integral}\index{q-Beta integral@$q$-Beta integral}
\frac{1}{2\cpi}\int_{-1}^1\frac{w(x)}{\sqrt{1-x^2}}\,dx=\frac{1}{2\cpi}\int_0^{\cpi}w(\cos@@{\theta})\,d\theta
=\frac{\qPochhammer{abcd}{q}{\infty}}{\qPochhammer{ab,ac,ad,bc,bd,cd,q}{q}{\infty}}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle w(x)=\frac{h(x,1)h(x,-1)h(x,q^{1/2})h(x,-q^{1/2})}{h(x,a)h(x,b)h(x,c)h(x,d)}
=\left|\frac{\qPochhammer{\expe^{2\iunit\theta}}{q}{\infty}}{\qPochhammer{a\expe^{\iunit\theta},b\expe^{\iunit\theta},c\expe^{\iunit\theta},d\expe^{\iunit\theta}}{q}{\infty}}\right|^2
=\frac{\qPochhammer{\expe^{2\iunit\theta},\expe^{-2\iunit\theta}}{q}{\infty}}
{\qPochhammer{a\expe^{\iunit\theta},a\expe^{-\iunit\theta},b\expe^{\iunit\theta},b\expe^{-\iunit\theta},c\expe^{\iunit\theta},c\expe^{-\iunit\theta},d\expe^{\iunit\theta},d\expe^{-\iunit\theta}}{q}{\infty}}}</math> &<br />
<math>{\displaystyle h(x,\alpha)=\prod_{k=0}^{\infty}\left(1-2\alpha xq^k+\alpha^2q^{2k}\right)
=\left|\qPochhammer{\alpha \expe^{\iunit\theta}}{q}{\infty}\right|^2
=\qPochhammer{\alpha \expe^{\iunit\theta},\alpha \expe^{-\iunit\theta}}{q}{\infty}}</math> &<br />
<math>{\displaystyle x=\cos@@{\theta}}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The q-binomial theorem and other summation formulas|The q-binomial theorem and other summation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|More integrals]] </div>
<div id="alignright"> [[Transformation formulas|Transformation formulas]] >> </div>
</div>

drmf_eof
drmf_bof
'''Transformation formulas'''
{{DISPLAYTITLE:Transformation formulas}}
<div id="drmf_head">
<div id="alignleft"> << [[More integrals|More integrals]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Transformation formulas]] </div>
<div id="alignright"> [[Some q-analogues of special functions|Some q-analogues of special functions]] >> </div>
</div>

== Transformation formulas ==

<math id="KLS:01.13:01">{\displaystyle
\index{Heine's transformation formula}\index{Transformation formula!Heine}
\qHyperrphis{2}{1}@@{a,b}{c}{q}{z}=\frac{\qPochhammer{az,b}{q}{\infty}}{\qPochhammer{c,z}{q}{\infty}}\
\qHyperrphis{2}{1}@@{b^{-1}c,z}{az}{q}{b}
}</math><br />
<math id="KLS:01.13:02">{\displaystyle
\index{Heine's transformation formula}\index{Transformation formula!Heine}
\qHyperrphis{2}{1}@@{a,b}{c}{q}{z}=\frac{\qPochhammer{b^{-1}c,bz}{q}{\infty}}{\qPochhammer{c,z}{q}{\infty}}\
\qHyperrphis{2}{1}@@{abc^{-1}z,b}{bz}{q}{\frac{c}{b}}\label{Heine2}
}</math><br />
<math id="KLS:01.13:03">{\displaystyle
\index{Heine's transformation formula}\index{Transformation formula!Heine}
\qHyperrphis{2}{1}@@{a,b}{c}{q}{z}=\frac{\qPochhammer{abc^{-1}z}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}\
\qHyperrphis{2}{1}@@{a^{-1}c,b^{-1}c}{c}{q}{\frac{abz}{c}}.\label{Heine3}
}</math><br />
<math id="KLS:01.13:04">{\displaystyle
\index{q-Euler transformation formula@$q$-Euler transformation formula}\index{Transformation formula!q-Euler@$q$-Euler}
\HyperpFq{2}{1}@@{a,b}{c}{z}=(1-z)^{c-a-b}\,\HyperpFq{2}{1}@@{c-a,c-b}{c}{z}
}</math><br />
<math id="KLS:01.13:05">{\displaystyle
\index{q-Pfaff-Kummer transformation formula@$q$-Pfaff-Kummer transformation formula}\index{Transformation formula!q-Pfaff-Kummer@$q$-Pfaff-Kummer}
\qHyperrphis{2}{1}@@{a,b}{c}{q}{z}=\frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}\
\qHyperrphis{2}{2}@@{a,b^{-1}c}{c,az}{q}{bz}
}</math><br />
<math id="KLS:01.13:06">{\displaystyle
\qHyperrphis{2}{1}@@{a,0}{c}{q}{z}
=\frac{\qPochhammer{az}{q}{\infty}}{\qPochhammer{c,z}{q}{\infty}}\,\qHyperrphis{1}{1}@@{z}{az}{q}{c}
}</math><br />
<math id="KLS:01.13:07">{\displaystyle
\qHyperrphis{2}{1}@@{a,0}{c}{q}{z}
=\frac{1}{\qPochhammer{z}{q}{\infty}}\,\qHyperrphis{1}{1}@@{a^{-1}c}{c}{q}{az},\label{Heinelimit2}
}</math><br />
<math id="KLS:01.13:08">{\displaystyle
\qHyperrphis{2}{1}@@{0,0}{c}{q}{z}
=\frac{1}{\qPochhammer{c,z}{q}{\infty}}\,\qHyperrphis{1}{1}@@{z}{0}{q}{c}
}</math><br />
<math id="KLS:01.13:09">{\displaystyle
\qHyperrphis{2}{1}@@{0,0}{c}{q}{z}
=\frac{1}{\qPochhammer{z}{q}{\infty}}\,\qHyperrphis{0}{1}@@{-}{c}{q}{cz},\label{Heinelimit4}
}</math><br />
<math id="KLS:01.13:10">{\displaystyle
\qHyperrphis{1}{1}@@{a}{c}{q}{z}
=\frac{\qPochhammer{a,z}{q}{\infty}}{\qPochhammer{c}{q}{\infty}}\,\qHyperrphis{2}{1}@@{a^{-1}c,0}{z}{q}{a}
}</math><br />
<math id="KLS:01.13:11">{\displaystyle
\qHyperrphis{1}{1}@@{a}{c}{q}{z}
=\qPochhammer{ac^{-1}z}{q}{\infty}\cdot\qHyperrphis{2}{1}@@{a^{-1}c,0}{c}{q}{\frac{az}{c}}\label{Heinelimit6}
}</math><br />
<math id="KLS:01.13:12">{\displaystyle
\qHyperrphis{2}{1}@@{a,b}{0}{q}{z}
=\frac{\qPochhammer{az,b}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}\,\qHyperrphis{2}{1}@@{z,0}{az}{q}{b}
}</math><br />
<math id="KLS:01.13:13">{\displaystyle
\qHyperrphis{2}{1}@@{a,b}{0}{q}{z}
=\frac{\qPochhammer{bz}{q}{\infty}}{\qPochhammer{z}{q}{\infty}}\,\qHyperrphis{1}{1}@@{b}{bz}{q}{az}.\label{Heinelimit8}
}</math><br />
<math id="KLS:01.13:14">{\displaystyle
\qHyperrphis{1}{1}@@{q^{-n}}{a}{q}{z}=\frac{(q^{-1}z)^n}{\qPochhammer{a}{q}{n}}\
\qHyperrphis{2}{1}@@{q^{-n},a^{-1}q^{1-n}}{0}{q}{\frac{aq^{n+1}}{z}}
}</math><br />
<math id="KLS:01.13:15">{\displaystyle
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{z}
=\frac{\qPochhammer{b}{q}{n}}{\qPochhammer{c}{q}{n}}q^{-n-\binomial{n}{2}}(-z)^n\
\qHyperrphis{2}{1}@@{q^{-n},c^{-1}q^{1-n}}{b^{-1}q^{1-n}}{q}{\frac{cq^{n+1}}{bz}}
}</math><br />
<math id="KLS:01.13:16">{\displaystyle
\qHyperrphis{2}{0}@@{q^{-n},b}{-}{q}{zq^n}=\qPochhammer{b}{q}{n}z^n\
\qHyperrphis{2}{1}@@{q^{-n},0}{b^{-1}q^{1-n}}{q}{\frac{q}{bz}}
}</math><br />
<math id="KLS:01.13:17">{\displaystyle
\index{Jackson's transformation formula}\index{Transformation formula!Jackson}
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{z}=\frac{\qPochhammer{bc^{-1}q^{-n}z}{q}{\infty}}
{\qPochhammer{bc^{-1}z}{q}{\infty}}\,\qHyperrphis{3}{2}@@{q^{-n},b^{-1}c,0}{c,b^{-1}cqz^{-1}}{q}{q}
}</math><br />
<math id="KLS:01.13:18">{\displaystyle
\qHyperrphis{3}{2}@@{q^{-n},a,0}{b,c}{q}{q}=\frac{\qPochhammer{b^{-1}q}{q}{\infty}}
{\qPochhammer{b^{-1}q^{1-n}}{q}{\infty}}\,\qHyperrphis{2}{1}@@{q^{-n},a^{-1}c}{c}{q}{\frac{aq}{b}}
}</math><br />
<math id="KLS:01.13:19">{\displaystyle
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{z}
{}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}\left(\frac{bz}{q}\right)^n\
\qHyperrphis{3}{2}@@{q^{-n},qz^{-1},c^{-1}q^{1-n}}{bc^{-1}q^{1-n},0}{q}{q}
{}
}</math><br />
<math id="KLS:01.13:20">{\displaystyle
\qHyperrphis{2}{1}@@{q^{-n},b}{c}{q}{z}
{}=\frac{\qPochhammer{b^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}\,\qHyperrphis{3}{2}@@{q^{-n},b,bc^{-1}q^{-n}z}{bc^{-1}q^{1-n},0}{q}{q}\label{Jackson4}
}</math><br />
<math id="KLS:01.13:21">{\displaystyle
\qHyperrphis{3}{2}@@{q^{-n},a,b}{c,0}{q}{q}=
\frac{\qPochhammer{b}{q}{n}}{\qPochhammer{c}{q}{n}}a^n\,\qHyperrphis{2}{1}@@{q^{-n},b^{-1}c}{b^{-1}q^{1-n}}{q}{\frac{q}{a}}
}</math><br />
<math id="KLS:01.13:22">{\displaystyle
\qHyperrphis{3}{2}@@{q^{-n},a,b}{c,0}{q}{q}=\frac{\qPochhammer{a^{-1}c}{q}{n}}{\qPochhammer{c}{q}{n}}a^n\,\qHyperrphis{2}{1}@@{q^{-n},a}{ac^{-1}q^{1-n}}{q}{\frac{bq}{c}}\label{Jackson6}
}</math><br />
<math id="KLS:01.13:23">{\displaystyle
\qHyperrphis{2}{0}@@{q^{-n},b}{-}{q}{z}=b^{-n}\
\qHyperrphis{3}{2}@@{q^{-n},b,bzq^{-n}}{0,0}{q}{q}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.13:24">{\displaystyle
\qHyperrphis{3}{2}@@{q^{-n},a,b}{0,0}{q}{q}=\qPochhammer{b}{q}{n}a^n\
\qHyperrphis{2}{1}@@{q^{-n},0}{b^{-1}q^{1-n}}{q}{\frac{q}{a}}
}</math><br />
<math id="KLS:01.13:25">{\displaystyle
\qHyperrphis{3}{2}@@{q^{-n},a,b}{0,0}{q}{q}=a^n\,\qHyperrphis{2}{0}@@{q^{-n},a}{-}{q}{\frac{bq^n}{a}}\label{Jacksonlimit3}
}</math><br />
<math id="KLS:01.13:26">{\displaystyle
\index{Sears' transformation formula}\index{Transformation formula!Sears}
\qHyperrphis{4}{3}@@{q^{-n},a,b,c}{d,e,f}{q}{q}
{}=\frac{\qPochhammer{a^{-1}e,a^{-1}f}{q}{n}}{\qPochhammer{e,f}{q}{n}}a^n\,\qHyperrphis{4}{3}@@{q^{-n},a,b^{-1}d,c^{-1}d}{d,a\expe^{-1}q^{1-n},af^{-1}q^{1-n}}{q}{q}
{}
}</math><br />
<math id="KLS:01.13:27">{\displaystyle
\index{Sears' transformation formula}\index{Transformation formula!Sears}
\qHyperrphis{4}{3}@@{q^{-n},a,b,c}{d,e,f}{q}{q}
{}=\frac{\qPochhammer{a,a^{-1}b^{-1}ef,a^{-1}c^{-1}ef}{q}{n}}{\qPochhammer{e,f,a^{-1}b^{-1}c^{-1}ef}{q}{n}}
{}\qHyperrphis{4}{3}@@{q^{-n},a^{-1}e,a^{-1}f,a^{-1}b^{-1}c^{-1}ef}{a^{-1}b^{-1}ef,a^{-1}c^{-1}ef,a^{-1}q^{1-n}}{q}{q},\label{Sears2}
}</math><br />
<math id="KLS:01.13:28">{\displaystyle
\index{Singh's transformation formula}\index{Transformation formula!Singh}
\qHyperrphis{4}{3}@@{a^2,b^2,c,d}{abq^{1/2},-abq^{1/2},-cd}{q}{q}=
\qHyperrphis{4}{3}\left(\genfrac{}{}{0pt}{}{a^2,b^2,c^2,d^2}{a^2b^2q,-cd,-cdq}\,;\,q^2,q^2\right)
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[More integrals|More integrals]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Transformation formulas]] </div>
<div id="alignright"> [[Some q-analogues of special functions|Some q-analogues of special functions]] >> </div>
</div>

drmf_eof
drmf_bof
'''Some q-analogues of special functions'''
{{DISPLAYTITLE:Some ''q''-analogues of special functions}}
<div id="drmf_head">
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Some q-analogues of special functions]] </div>
<div id="alignright"> [[The q-derivative and q-integral|The q-derivative and q-integral]] >> </div>
</div>

== Some ''q''-analogues of special functions ==

<math id="KLS:01.14:01">{\displaystyle
\index{q-Exponential function@$q$-Exponential function}
\qexpKLS{q}@{z}:=\qHyperrphis{1}{0}@@{0}{-}{q}{z}=\sum_{n=0}^{\infty}\frac{z^n}{\qPochhammer{q}{q}{n}}
=\frac{1}{\qPochhammer{z}{q}{\infty}},\quad 0<|q|<1
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|<1}</math></div><br />
<math id="KLS:01.14:02">{\displaystyle
\qExpKLS{q}@{z}:=\qHyperrphis{0}{0}@@{-}{-}{q}{-z}=
\sum_{n=0}^{\infty}\frac{q^{\binomial{n}{2}}}{\qPochhammer{q}{q}{n}}z^n=\qPochhammer{-z}{q}{\infty}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<|q|<1}</math></div><br />
<math id="KLS:01.14:03">{\displaystyle
\qexpKLS{q}@{z)\E_q(-z}=1
}</math><br />
<math id="KLS:01.14:04">{\displaystyle
\lim\limits_{q\rightarrow 1}\qexpKLS{q}@{(1-q)z}=\lim\limits_{q\rightarrow 1}\qExpKLS{q}@{(1-q)z}=\expe^z
}</math><br />
<math id="KLS:01.14:05">{\displaystyle
\qcosKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)+{\mathrm e}_q(-\iunit z}}{2}=
\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n}}{\qPochhammer{q}{q}{2n}}
}</math><br />
<math id="KLS:01.14:06">{\displaystyle
\qsinKLS{q}@{z}:=\frac{\qexpKLS{q}@{\iunit z)-{\mathrm e}_q(-\iunit z}}{2\iunit}=
\sum_{n=0}^{\infty}\frac{(-1)^nz^{2n+1}}{\qPochhammer{q}{q}{2n+1}}
}</math><br />
<math id="KLS:01.14:07">{\displaystyle
\qCosKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}+\qExpKLS{q}@{-\iunit z}}{2}
=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n}{2}}z^{2n}}{\qPochhammer{q}{q}{2n}}
}</math><br />
<math id="KLS:01.14:08">{\displaystyle
\qSinKLS{q}@{z}:=\frac{\qExpKLS{q}@{\iunit z}-\qExpKLS{q}@{-\iunit z}}{2\iunit}
=\sum_{n=0}^{\infty}\frac{(-1)^nq^{\binomial{2n+1}{2}}z^{2n+1}}{\qPochhammer{q}{q}{2n+1}}
}</math><br />
<math id="KLS:01.14:09">{\displaystyle
\qexpKLS{q}@{\iunit z}=\qcosKLS{q}@{z}+\iunit\qsinKLS{q}@{z}\quad\textrm{and}\quad \qExpKLS{q}@{\iunit z}=\qCosKLS{q}@{z}+\iunit\:\qSinKLS{q}@{z}
}</math><br />
<math id="KLS:01.14:10">{\displaystyle
\qcosKLS{q}@{z}\qCosKLS{q}@{z}+\qsinKLS{q}@{z}\qSinKLS{q}@{z}=1
}</math><br />
<math id="KLS:01.14:11">{\displaystyle
\qsinKLS{q}@{z}\qCosKLS{q}@{z}-\qcosKLS{q}@{z}\qSinKLS{q}@{z}=0
}</math><br />
<math id="KLS:01.14:12">{\displaystyle
\index{q-Bessel function@$q$-Bessel function}\index{Jackson's q-Bessel function@Jackson's $q$-Bessel function}
\JacksonqBesselI{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}
\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{2}{1}@@{0,0}{q^{\nu+1}}{q}{-\frac{z^2}{4}}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|<2}</math></div><br />
<math id="KLS:01.14:13">{\displaystyle
\JacksonqBesselII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}
\left(\frac{z}{2}\right)^{\nu}\,\qHyperrphis{0}{1}@@{-}{q^{\nu+1}}{q}{-\frac{q^{\nu+1}z^2}{4}}
}</math><br />
<math id="KLS:01.14:14">{\displaystyle
\JacksonqBesselII{\nu}@{z}{q}=\qPochhammer{-\frac{z^2}{4}}{q}{\infty}\cdot \JacksonqBesselI{\nu}@{z}{q}
}</math>
<div align="right">Constraint(s): <math>{\displaystyle |z|<2}</math></div><br />
<math id="KLS:01.14:15">{\displaystyle
\lim\limits_{q\rightarrow 1}J_{\nu}^{(k)}((1-q)z;q)=\BesselJ{\nu}@{z}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle k=1,2}</math></div><br />
<math id="KLS:01.14:16">{\displaystyle
\JacksonqBesselIII{\nu}@{z}{q}:=\frac{\qPochhammer{q^{\nu+1}}{q}{\infty}}{\qPochhammer{q}{q}{\infty}}
z^{\nu}\,\qHyperrphis{1}{1}@@{0}{q^{\nu+1}}{q}{qz^2}
}</math><br />
<math id="KLS:01.14:17">{\displaystyle
\lim\limits_{q\rightarrow 1}\JacksonqBesselIII{\nu}@{(1-q)z}{q}=\BesselJ{\nu}@{2z}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Transformation formulas|Transformation formulas]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Some q-analogues of special functions]] </div>
<div id="alignright"> [[The q-derivative and q-integral|The q-derivative and q-integral]] >> </div>
</div>

drmf_eof
drmf_bof
'''The q-derivative and q-integral'''
{{DISPLAYTITLE:The ''q''-derivative and ''q''-integral}}
<div id="drmf_head">
<div id="alignleft"> << [[Some q-analogues of special functions|Some q-analogues of special functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-derivative and q-integral]] </div>
<div id="alignright"> [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] >> </div>
</div>

== The ''q''-derivative and ''q''-integral ==

<math id="KLS:01.15:01">{\displaystyle
\index{q-Derivative operator@$q$-Derivative operator}
\qderiv{q}f(z):=\left\{\begin{array}{ll}
\displaystyle \frac{f(z)-f(qz)}{(1-q)z}, & z\neq 0\\[5mm]
f'(0), & z=0.\end{array}\right.
}</math><br />
<math id="KLS:01.15:02">{\displaystyle
\qderiv[0]{q}f:=f\quad\textrm{and}\quad
\qderiv[n]{q}f:=\qderiv{q}@{\qderiv[n-1]{q}f}
\quad n=1,2,3,\ldots
}</math><br />
<math id="KLS:01.15:03">{\displaystyle
\lim\limits_{q\rightarrow 1}\qderiv{q}f(z)=f'(z)
}</math><br />
<math id="KLS:01.15:04">{\displaystyle
\qderiv{q}@{f(\gamma x)}=\gamma\left(\qderiv{q}f\right)(\gamma x)
}</math><br />
<math id="KLS:01.15:05">{\displaystyle
\qderiv[n]{q}@{f(\gamma x)}=\gamma^n\left(\qderiv[n]{q}f\right)(\gamma x)
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.15:06">{\displaystyle
\qderiv{q}@{f(x)g(x)}=
f(qx)\qderiv{q}g(x)+g(x)\qderiv{q}f(x)
}</math><br />
<math id="KLS:01.15:07">{\displaystyle
\qderiv[n]{q}@{f(x)g(x)}=\sum_{k=0}^n\qBinomial{n}{k}{q}
\left(\qderiv[n-k]{q}f\right)(q^kx)\left(\qderiv[k]{q}g\right)(x)
}</math>
<div align="right">Constraint(s): <math>{\displaystyle n=0,1,2,\ldots}</math></div><br />
<math id="KLS:01.15:08">{\displaystyle
\index{q-Integral@$q$-Integral}\index{Jackson-Thomae q-integral@Jackson-Thomae $q$-integral}
\int_0^zf(t)\,d_qt:=z(1-q)\sum_{n=0}^{\infty}f(q^nz)q^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.15:09">{\displaystyle
\int_0^{\infty}f(t)\,d_qt:=(1-q)\sum_{n=-\infty}^{\infty}f(q^n)q^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.15:10">{\displaystyle
\lim\limits_{q\rightarrow 1}\int_0^zf(t)\,d_qt=\int_0^zf(t)\,dt
}</math><br />
<math id="KLS:01.15:11">{\displaystyle
\int_a^bf(t)\,d_qt=b(1-q)\sum_{n=0}^{\infty}f(bq^n)q^n-a(1-q)\sum_{n=0}^{\infty}f(aq^n)q^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.15:12">{\displaystyle
\int_{-\infty}^{\infty}f(t)\,d_qt=(1-q)\sum_{n=-\infty}^{\infty}\left\{f(q^n)+f(-q^n)\right\}q^n
}</math>
<div align="right">Constraint(s): <math>{\displaystyle 0<q<1}</math></div><br />
<math id="KLS:01.15:13">{\displaystyle
F(z)=\int_0^zf(t)\,d_qt\quad\Longrightarrow\quad\qderiv{q}F(z)=f(z)
}</math><br />
<math id="KLS:01.15:14">{\displaystyle
\int_a^b\qderiv{q}f(t)\,d_qt=f(b)-f(a)
}</math><br />
<math id="KLS:01.15:15">{\displaystyle
\int_a^bg(x)\qderiv{q}f(x)\,d_qx=\Big[f(x)g(x)\Big]_a^b-\int_a^bf(qx)\qderiv{q}g(x)\,d_qx
}</math><br />
<math id="KLS:01.15:16">{\displaystyle
\int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt,ct}{q}{\infty}}{\qPochhammer{dt,et,ft}{q}{\infty}}\,d_qt
{}=(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,cd^{-1},c\expe^{-1},cf^{-1}}{q}{\infty}}
{\qPochhammer{ad,ae,af,bd,be,bf}{q}{\infty}}
}</math><br />
<math id="KLS:01.15:17">{\displaystyle
\int_a^b\frac{\qPochhammer{a^{-1}qt,b^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt
=(b-a)(1-q)\,\frac{\qPochhammer{q,a^{-1}bq,ab^{-1}q,abde}{q}{\infty}}{\qPochhammer{ad,ae,bd,be}{q}{\infty}}
}</math><br />
<math id="KLS:01.15:18">{\displaystyle
\int_a^{\infty}\frac{\qPochhammer{a^{-1}qt}{q}{\infty}}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt
=(1-q)\,\frac{\qPochhammer{q,a,a^{-1}q,ade,a^{-1}d^{-1}\expe^{-1}q}{q}{\infty}}
{\qPochhammer{ad,ae,d,d^{-1}q,e,\expe^{-1}q}{q}{\infty}}
}</math><br />
<math id="KLS:01.15:19">{\displaystyle
\int_{-\infty}^{\infty}\frac{1}{\qPochhammer{dt,et}{q}{\infty}}\,d_qt
{}=(1-q)\,\frac{\qPochhammer{q,-q,-1,-de,-d^{-1}\expe^{-1}q}{q}{\infty}}
{\qPochhammer{d,-d,d^{-1}q,-d^{-1}q,e,-e,\expe^{-1}q,-\expe^{-1}q}{q}{\infty}}
}</math><br />
<div id="drmf_foot">
<div id="alignleft"> << [[Some q-analogues of special functions|Some q-analogues of special functions]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|The q-derivative and q-integral]] </div>
<div id="alignright"> [[Shift operators and Rodrigues-type formulas|Shift operators and Rodrigues-type formulas]] >> </div>
</div>

drmf_eof
drmf_bof
'''Shift operators and Rodrigues-type formulas'''
{{DISPLAYTITLE:Shift operators and Rodrigues-type formulas}}
<div id="drmf_head">
<div id="alignleft"> << [[The q-derivative and q-integral|The q-derivative and q-integral]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Shift operators and Rodrigues-type formulas]] </div>
<div id="alignright"> [[Orthogonal Polynomials|Orthogonal Polynomials]] >> </div>
</div>

== Shift operators and Rodrigues-type formulas ==

<math id="KLS:01.16:01">{\displaystyle
f'(x)=\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{df(x)}{dx}
}</math><br />
<math id="KLS:01.16:02">{\displaystyle
\Delta f(x)=f(x+1)-f(x)
}</math><br />
<math id="KLS:01.16:03">{\displaystyle
\nabla f(x)=f(x)-f(x-1)
}</math><br />
<math id="KLS:01.16:04">{\displaystyle
\delta f(x)=f(x+\textstyle\frac{1}{2}\iunit)-f(x-\textstyle\frac{1}{2}\iunit)
}</math><br />
<math id="KLS:01.16:05">{\displaystyle
\delta^2 f(x)=f(x+\iunit)-f(x)-f(x)+f(x-\iunit)=f(x+\iunit)-2f(x)+f(x-\iunit)
}</math><br />
<math id="KLS:01.16:06">{\displaystyle
\delta x=x+\textstyle\frac{1}{2}\iunit-(x-\textstyle\frac{1}{2}\iunit)=\iunit
\quad\textrm{and}\quad
\delta x^2=(x+\textstyle\frac{1}{2}\iunit)^2-(x-\textstyle\frac{1}{2}\iunit)^2=2\iunit x
}</math><br />
<math id="KLS:01.16:07">{\displaystyle
\Delta\lambda(x)=2x+\gamma+\delta+2\quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta
}</math><br />
<math id="KLS:01.16:08">{\displaystyle
\Delta\mu(x)=q^{-x-1}(1-q)(1-\gamma\delta q^{2x+2})\quad\textrm{and}\quad
\nabla\mu(x)=q^{-x}(1-q)(1-\gamma\delta q^{2x})
}</math><br />
<math id="KLS:01.16:09">{\displaystyle
\Delta\lambda(x)=q^{-x-1}(1-q)(1-cq^{2x-N+1})\quad\textrm{and}\quad
\nabla\lambda(x)=q^{-x}(1-q)(1-cq^{2x-N-1})
}</math><br />
<math id="KLS:01.16:10">{\displaystyle
\Delta q^{-x}=q^{-x-1}(1-q)\quad\textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q)
}</math><br />
<math id="KLS:01.16:11">{\displaystyle
D_qf(x):=\frac{\delta_qf(x)}{\delta_qx}
}</math>
<div align="right">Substitution(s): <math>{\displaystyle x=\cos@@{\theta}}</math></div><br />
<math id="KLS:01.16:12">{\displaystyle
\delta_qf(\expe^{\iunit\theta})
=f(q^{\frac{1}{2}}\expe^{\iunit\theta})-f(q^{-\frac{1}{2}}\expe^{\iunit\theta})
}</math><br />
<math id="KLS:01.16:13">{\displaystyle
\delta_qx=-\textstyle\frac{1}{2}q^{-\frac{1}{2}}(1-q)(\expe^{\iunit\theta}-\expe^{-\iunit\theta})
}</math>
<div align="right">Substitution(s): <math>{\displaystyle x=\cos@@{\theta}}</math></div><br />
<div id="drmf_foot">
<div id="alignleft"> << [[The q-derivative and q-integral|The q-derivative and q-integral]] </div>
<div id="aligncenter"> [[Orthogonal_Polynomials#Sections_in_KLS_Chapter_1|Shift operators and Rodrigues-type formulas]] </div>
<div id="alignright"> [[Orthogonal Polynomials|Orthogonal Polynomials]] >> </div>
</div>

drmf_eof
