shift of variable

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1: 33.25 Approximations

§33.25 Approximations

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2: 33.23 Methods of Computation

§33.23 Methods of Computation

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3: 28.20 Definitions and Basic Properties

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§28.20(vii) Shift of Variable

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4: 22.4 Periods, Poles, and Zeros

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Table 22.4.3: Half- or quarter-period shifts of variable for the Jacobian elliptic functions.

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	$u$
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5: 26.15 Permutations: Matrix Notation

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26.15.11 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=\prod _{j=1}^n(x+b_j-j+1).$$

ⓘ

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $j$: nonnegative integer, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E11

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

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26.15.12 $$\sum _{k=0}^n{r}_{n-k}(B){\left(x-k+1\right)}_k=x^n,$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $x$: real variable, $k$: nonnegative integer, $n$: nonnegative integer, $B$: subset and $r_j(B)$: number of rook positions

Permalink:

http://dlmf.nist.gov/26.15.E12

Encodings:

pMML, png, TeX

Errata (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §26.15, §26.15 and Ch.26

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6: 31.11 Expansions in Series of Hypergeometric Functions

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31.11.3_1 $$P_j^5=\frac{{\left(\lambda \right)}_j{\left(1-\gamma +\lambda \right)}_j}{{\left(1+\lambda -\mu \right)}_{2j}}{z}^{-\lambda -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\lambda +j,1-\gamma +\lambda +j}{1+\lambda -\mu +2j};\frac{1}{z}),$$

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Defines:

$P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.3_2 $$P_j^6=\frac{{\left(\lambda -\mu \right)}_{2j}}{{\left(1-\mu \right)}_j{\left(\gamma -\mu \right)}_j}{z}^{-\mu +j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\mu -j,1-\gamma +\mu -j}{1-\lambda +\mu -2j};\frac{1}{z}).$$

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Defines:

$P_j^6$: second Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation (locally)

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $\lambda$ and $\mu$

Referenced by:

§31.11(ii), Erratum (V1.1.7) for Additions

Permalink:

http://dlmf.nist.gov/31.11.E3_2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.7):

This equation was added.

Suggested 2022-09-14 by Hans Volkmer

See also:

Annotations for §31.11(ii), §31.11 and Ch.31

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31.11.12 $$P_j^5=\frac{{\left(\alpha \right)}_j{\left(1-\gamma +\alpha \right)}_j}{{\left(1+\alpha -\beta +ϵ\right)}_{2j}}{z}^{-\alpha -j}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha +j,1-\gamma +\alpha +j}{1+\alpha -\beta +ϵ+2j};\frac{1}{z}),$$

ⓘ

Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $\gamma$: real or complex parameter, $ϵ$: real or complex parameter, $j$: nonnegative integer, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $P_j^5$: first Kummer solution at $\mathrm{\infty }$ of generalized hypergeometric differential equation

Referenced by:

§31.11(iii), §31.11(iii), §31.11(iii), Erratum (V1.1.7) for Subsection 31.11(iii)

Permalink:

http://dlmf.nist.gov/31.11.E12

Encodings:

pMML, png, TeX

Notational Change (effective with 1.1.7):

We have rewritten the gamma functions in the prefactor more concisely using Pochhammer symbols.

See also:

Annotations for §31.11(iii), §31.11 and Ch.31

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7: 33.2 Definitions and Basic Properties

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33.2.7 $$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )={(\mp \mathrm{i})}^{\mathrm{\ell }}{\mathrm{e}}^{(\pi \eta /2)\pm \mathrm{i}{\sigma }_{\mathrm{\ell }}\left(\eta \right)}{W}_{\mp \mathrm{i}\eta ,\mathrm{\ell }+\frac{1}{2}}\left(\mp 2\mathrm{i}\rho \right),$$

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Defines:

$H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

Referenced by:

§13.18(vi), §33.23(vi)

Permalink:

http://dlmf.nist.gov/33.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

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33.2.9 $${\theta }_{\mathrm{\ell }}(\eta ,\rho )=\rho -\eta \ln \left(2\rho \right)-\frac{1}{2}\mathrm{\ell }\pi +{\sigma }_{\mathrm{\ell }}\left(\eta \right),$$

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Defines:

${\theta }_{\mathrm{\ell }}(\eta ,\rho )$: phase of Coulomb functions

Symbols:

${\sigma }_{\mathrm{\ell }}\left(\eta \right)$: Coulomb phase shift, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: principal branch of logarithm function, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable and $\eta$: real parameter

A&S Ref:

14.5.5

Referenced by:

§33.10(i), §33.11

Permalink:

http://dlmf.nist.gov/33.2.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §33.2(iii), §33.2 and Ch.33

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8: 18.12 Generating Functions

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18.12.2_5 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\alpha +1};\frac{1-R-z}{2}){}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\gamma ,\alpha +\beta +1-\gamma }{\beta +1};\frac{1-R+z}{2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\gamma \right)}_n{\left(\alpha +\beta +1-\gamma \right)}_n}{{\left(\alpha +1\right)}_n{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ $R=\sqrt{1-2xz+z^2}$, ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proved:

Rainville (1960, §141, (2)); Brafman’s generating function(proved)

Source:

Koekoek et al. (2010, (9.8.15))

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) §18.12

Permalink:

http://dlmf.nist.gov/18.12.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3 $${(1+z)}^{-\alpha -\beta -1}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}(\alpha +\beta +1),\frac{1}{2}(\alpha +\beta +2)}{\beta +1};\frac{2(x+1)z}{{(1+z)}^2})=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\alpha +\beta +1\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\alpha ,\beta )}\left(x\right)z^n,$$ ,

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Referenced by:

(18.12.3_5), §18.12, §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.3_5 $$\frac{1+z}{{(1-2xz+z^2)}^{\beta +\frac{3}{2}}}=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\beta +2\right)}_n}{{\left(\beta +1\right)}_n}{P}_n^{(\beta +1,\beta )}\left(x\right)z^n,$$ ,

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Proof sketch:

Substitute $\alpha =\beta +1$ in (18.12.3) and use (15.4.6).

Referenced by:

§18.12, §18.12, Erratum (V1.2.0) §18.12

Permalink:

http://dlmf.nist.gov/18.12.E3_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.4 $${(1-2xz+z^2)}^{-\lambda }=\sum _{n=0}^{\mathrm{\infty }}{C}_n^{(\lambda )}\left(x\right)z^n=\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(2\lambda \right)}_n}{{\left(\lambda +\frac{1}{2}\right)}_n}{P}_n^{(\lambda -\frac{1}{2},\lambda -\frac{1}{2})}\left(x\right)z^n,$$ .

ⓘ

Symbols:

$P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.3

Referenced by:

§18.12, §18.18(viii)

Permalink:

http://dlmf.nist.gov/18.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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18.12.14 $$\mathrm{\Gamma }\left(\alpha +1\right){(xz)}^{-\frac{1}{2}\alpha }{\mathrm{e}}^z{J}_{\alpha }\left(2\sqrt{xz}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(\alpha )}\left(x\right)}{{\left(\alpha +1\right)}_n}{z}^n.$$

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

A&S Ref:

22.9.16

Referenced by:

§18.12

Permalink:

http://dlmf.nist.gov/18.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.12, §18.12 and Ch.18

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9: 5.18 $q$-Gamma and $q$-Beta Functions

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5.18.1 $${(a;q)}_n=\prod _{k=0}^{n-1}(1-a{q}^k),$$ $n=0,1,2,\mathrm{\dots }$,

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Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $n$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

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5.18.2 $$n{!}_q=1(1+q)\mathrm{\cdots }(1+q+\mathrm{\cdots }+q^{n-1})={(q;q)}_n{(1-q)}^{-n}.$$

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Defines:

${!}_q$: $q$-factorial (as in $n{!}_q$)

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.18.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

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5.18.3 $${(a;q)}_{\mathrm{\infty }}=\prod _{k=0}^{\mathrm{\infty }}(1-a{q}^k).$$

ⓘ

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable, $k$: nonnegative integer and $a$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(i), §5.18 and Ch.5

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5.18.4 $${\mathrm{\Gamma }}_q\left(z\right)={(q;q)}_{\mathrm{\infty }}{(1-q)}^{1-z}/{(q^z;q)}_{\mathrm{\infty }},$$

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Defines:

${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real or complex variable and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.18.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(ii), §5.18 and Ch.5

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5.18.12 $${\mathrm{B}}_q(a,b)={\int }_0^1\frac{t^{a-1}{(tq;q)}_{\mathrm{\infty }}}{{(t{q}^b;q)}_{\mathrm{\infty }}}{d}_{q}t,$$ , $\mathrm{\Re }a>0$, $\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\int$: integral, ${\mathrm{B}}_q(a,b)$: $q$-beta function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, $\mathrm{\Re }$: real part, $q$: real or complex variable, $a$: real or complex variable and $b$: real or complex variable

Permalink:

http://dlmf.nist.gov/5.18.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §5.18(iii), §5.18 and Ch.5

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10: 18.27 $q$-Hahn Class

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18.27.14 $$\sum _{y=0}^{\mathrm{\infty }}{p}_n(q^y)p_m(q^y)\frac{{(bq;q)}_y{(aq)}^y}{{(q;q)}_y}=h_n{\delta }_{n,m},$$ ,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $y$: real variable, $p_n(x)$: polynomial of degree $n$ , $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iv), §18.27 and Ch.18

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18.27.14_3 $$\underset{c\uparrow 0}{lim}{P}_n(bqx;b,a,c;q)={(-b)}^n{q}^{n(n+1)/2}\frac{{(qa;q)}_n}{{(qb;q)}_n}{p}_n(x;a,b;q).$$

ⓘ

Symbols:

$P_n(x;a,b,c;q)$: big $q$-Jacobi polynomial, $p_n(x;a,b;q)$: little $q$-Jacobi polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Proved:

Koornwinder (1994, (4.45),(4.46)); Combine with (18.27.5).(proved)

Referenced by:

§18.27(iv)

Permalink:

http://dlmf.nist.gov/18.27.E14_3

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(iv), §18.27(iv), §18.27 and Ch.18

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18.27.16 $${\int }_0^{\mathrm{\infty }}{L}_n^{(\alpha )}(x;q)L_m^{(\alpha )}(x;q)\frac{x^{\alpha }}{{(-x;q)}_{\mathrm{\infty }}}dx=\frac{{(q^{\alpha +1};q)}_n}{{(q;q)}_n{q}^n}{h}_0^{(1)}{\delta }_{n,m},$$ $\alpha >-1$,

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $L_n^{(\alpha )}(x;q)$: $q$-Laguerre polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer, $x$: real variable and $h_n$

Permalink:

http://dlmf.nist.gov/18.27.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §18.27(v), §18.27 and Ch.18

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18.27.17_1 $$h_0^{(1)}=\frac{{(q^{-\alpha };q)}_{\mathrm{\infty }}}{{(q;q)}_{\mathrm{\infty }}}\mathrm{\Gamma }\left(\alpha +1\right)\mathrm{\Gamma }\left(-\alpha \right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable and $h_n$

Source:

Koekoek et al. (2010, (14.21.2))

Referenced by:

§18.27(v), Erratum (V1.2.0) §18.27

Permalink:

http://dlmf.nist.gov/18.27.E17_1

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(v), §18.27 and Ch.18

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18.27.20_5 $$\underset{q\to 1}{lim}\frac{{(q;q)}_n{S}_n(q^{-1}x\sqrt{2(1-q)}+1;q)}{{\left(\frac{1-q}{2}\right)}^{n/2}}={(-1)}^n{H}_n\left(x\right).$$

ⓘ

Symbols:

$H_n\left(x\right)$: Hermite polynomial, $S_n(x;q)$: Stieltjes–Wigert polynomial, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $n$: nonnegative integer and $x$: real variable

Source:

Koekoek et al. (2010, (14.27.14))

Referenced by:

§18.27(vi), Erratum (V1.2.0) §18.27

Permalink:

http://dlmf.nist.gov/18.27.E20_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.27(vi), §18.27(vi), §18.27 and Ch.18

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