# shift of variable

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## 1—10 of 135 matching pages

##### 6: 31.11 Expansions in Series of Hypergeometric Functions
31.11.3_1 $P_{j}^{5}=\frac{{\left(\lambda\right)_{j}}{\left(1-\gamma+\lambda\right)_{j}}}% {{\left(1+\lambda-\mu\right)_{2j}}}z^{-\lambda-j}\*{{}_{2}F_{1}}\left({\lambda% +j,1-\gamma+\lambda+j\atop 1+\lambda-\mu+2j};\frac{1}{z}\right),$
31.11.3_2 $P_{j}^{6}=\frac{{\left(\lambda-\mu\right)_{2j}}}{{\left(1-\mu\right)_{j}}{% \left(\gamma-\mu\right)_{j}}}z^{-\mu+j}\*{{}_{2}F_{1}}\left({\mu-j,1-\gamma+% \mu-j\atop 1-\lambda+\mu-2j};\frac{1}{z}\right).$
31.11.12 $P_{j}^{5}=\frac{{\left(\alpha\right)_{j}}{\left(1-\gamma+\alpha\right)_{j}}}{{% \left(1+\alpha-\beta+\epsilon\right)_{2j}}}z^{-\alpha-j}\*{{}_{2}F_{1}}\left({% \alpha+j,1-\gamma+\alpha+j\atop 1+\alpha-\beta+\epsilon+2j};\frac{1}{z}\right),$
##### 7: 33.2 Definitions and Basic Properties
33.2.9 ${\theta_{\ell}}\left(\eta,\rho\right)=\rho-\eta\ln\left(2\rho\right)-\tfrac{1}% {2}\ell\pi+{\sigma_{\ell}}\left(\eta\right),$
##### 8: 18.12 Generating Functions
18.12.2_5 ${{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\alpha+1};\frac{1-R-z}{2}% \right)\*{{}_{2}F_{1}}\left({\gamma,\alpha+\beta+1-\gamma\atop\beta+1};\frac{1% -R+z}{2}\right)=\sum_{n=0}^{\infty}\frac{{\left(\gamma\right)_{n}}{\left(% \alpha+\beta+1-\gamma\right)_{n}}}{{\left(\alpha+1\right)_{n}}{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $R=\sqrt{1-2xz+z^{2}}$, $|z|<1$,
18.12.3 $(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({\tfrac{1}{2}(\alpha+\beta+1),% \tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2(x+1)z}{(1+z)^{2}}\right)=% \sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right)_{n}}}{{\left(\beta+1% \right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},$ $|z|<1$,
18.12.3_5 $\frac{1+z}{(1-2xz+z^{2})^{\beta+\frac{3}{2}}}=\sum_{n=0}^{\infty}\frac{{\left(% 2\beta+2\right)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\beta+1,\beta)}_{n}\left(% x\right)z^{n},$ $|z|<1$,
18.12.4 $(1-2xz+z^{2})^{-\lambda}=\sum_{n=0}^{\infty}C^{(\lambda)}_{n}\left(x\right)z^{% n}=\sum_{n=0}^{\infty}\frac{{\left(2\lambda\right)_{n}}}{{\left(\lambda+\tfrac% {1}{2}\right)_{n}}}P^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}_{n}\left(x% \right)z^{n},$ $|z|<1$.
18.12.14 $\Gamma\left(\alpha+1\right)(xz)^{-\frac{1}{2}\alpha}{\mathrm{e}}^{z}J_{\alpha}% \left(2\sqrt{xz}\right)=\sum_{n=0}^{\infty}\frac{L^{(\alpha)}_{n}\left(x\right% )}{{\left(\alpha+1\right)_{n}}}z^{n}.$
##### 9: 5.18 $q$-Gamma and $q$-Beta Functions
5.18.1 $\left(a;q\right)_{n}=\prod_{k=0}^{n-1}(1-aq^{k}),$ $n=0,1,2,\dots$,
5.18.2 $n!_{q}=1(1+q)\cdots(1+q+\dots+q^{n-1})=\left(q;q\right)_{n}(1-q)^{-n}.$
5.18.3 $\left(a;q\right)_{\infty}=\prod_{k=0}^{\infty}(1-aq^{k}).$
5.18.4 $\Gamma_{q}\left(z\right)=\left(q;q\right)_{\infty}(1-q)^{1-z}/\left(q^{z};q% \right)_{\infty},$
5.18.12 $\mathrm{B}_{q}\left(a,b\right)=\int_{0}^{1}\frac{t^{a-1}\left(tq;q\right)_{% \infty}}{\left(tq^{b};q\right)_{\infty}}\,{\mathrm{d}}_{q}t,$ $0, $\Re a>0$, $\Re b>0$.
##### 10: 18.27 $q$-Hahn Class
18.27.14 $\sum_{y=0}^{\infty}p_{n}(q^{y})p_{m}(q^{y})\frac{\left(bq;q\right)_{y}(aq)^{y}% }{\left(q;q\right)_{y}}=h_{n}\delta_{n,m},$ $0,
18.27.14_3 $\lim_{c\uparrow 0}P_{n}\left(bqx;b,a,c;q\right)=(-b)^{n}q^{\ifrac{n(n+1)}{2}}% \frac{\left(qa;q\right)_{n}}{\left(qb;q\right)_{n}}p_{n}\left(x;a,b;q\right).$
18.27.16 $\int_{0}^{\infty}L^{(\alpha)}_{n}\left(x;q\right)L^{(\alpha)}_{m}\left(x;q% \right)\frac{x^{\alpha}}{\left(-x;q\right)_{\infty}}\,\mathrm{d}x=\frac{\left(% q^{\alpha+1};q\right)_{n}}{\left(q;q\right)_{n}q^{n}}h_{0}^{(1)}\delta_{n,m},$ $\alpha>-1$,
18.27.17_1 $h_{0}^{(1)}=\frac{\left(q^{-\alpha};q\right)_{\infty}}{\left(q;q\right)_{% \infty}}\Gamma\left(\alpha+1\right)\Gamma\left(-\alpha\right),$