Pochhammer integral

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11: 25.11 Hurwitz Zeta Function

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25.11.28 $$\zeta (s,a)=\frac{1}{2}{a}^{-s}+\frac{a^{1-s}}{s-1}+\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}+\frac{1}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\left(\frac{1}{{\mathrm{e}}^x-1}-\frac{1}{x}+\frac{1}{2}-\sum _{k=1}^n\frac{B_{2k}}{(2k)!}{x}^{2k-1}\right)x^{s-1}{\mathrm{e}}^{-ax}dx,$$ $\mathrm{\Re }s>-(2n+1)$, $s\ne 1$, $\mathrm{\Re }a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable

Keywords:

improper integral, integral representation

Proof sketch:

Derivable from (25.11.27) by adding and subtracting ${\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that ${({\mathrm{e}}^x-1)}^{-1}-x^{-1}+1/2-{\sum }_{k=1}^n{B}_{2k}{x}^{2k-1}/(2k)!=O\left(x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence.

Referenced by:

Erratum (V1.0.9) for Chapters 7, 25

Permalink:

http://dlmf.nist.gov/25.11.E28

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.9):

We have rewritten the original summation ${\sum }_{k=1}^n\frac{\mathrm{\Gamma }\left(s+2k-1\right)}{\mathrm{\Gamma }\left(s\right)}\frac{B_{2k}}{(2k)!}{a}^{-2k-s+1}$ more concisely as ${\sum }_{k=1}^n\frac{B_{2k}}{(2k)!}{\left(s\right)}_{2k-1}{a}^{1-s-2k}$ using the Pochhammer symbol.

See also:

Annotations for §25.11(vii), §25.11 and Ch.25

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12: 2.6 Distributional Methods

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2.6.13 $$⟨t^{-s-\alpha },\varphi ⟩=\frac{1}{{\left(\alpha \right)}_s}{\int }_0^{\mathrm{\infty }}{t}^{-\alpha }{\varphi }^{(s)}(t)dt,$$ $\varphi \in 𝒯$,

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $\in$: element of, $\int$: integral and $𝒯$: space of decreasing functions

Referenced by:

§2.6(ii), §2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §2.6(ii), §2.6 and Ch.2

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13: 14.12 Integral Representations

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14.12.7 $$P_{\nu }^m\left(x\right)=\frac{{\left(\nu +1\right)}_m}{\pi }{\int }_0^{\pi }{\left(x+{\left(x^2-1\right)}^{1/2}\cos \varphi \right)}^{\nu }\cos \left(m\varphi \right)d\varphi ,$$

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Symbols:

$P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer and $\nu$: general degree

Permalink:

http://dlmf.nist.gov/14.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §14.12(ii), §14.12 and Ch.14

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14: 17.6 ${}_2{}^{}\varphi _1^{}$ Function

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17.6.28 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{q^{\alpha },q^{\beta }}{q^{\gamma }};q,z)=\frac{{\mathrm{\Gamma }}_q\left(\gamma \right)}{{\mathrm{\Gamma }}_q\left(\beta \right){\mathrm{\Gamma }}_q\left(\gamma -\beta \right)}{\int }_0^1\frac{t^{\beta -1}{(tq;q)}_{\gamma -\beta -1}}{{(xt;q)}_{\alpha }}{d}_{q}t.$$

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Symbols:

$\int$: integral, ${\mathrm{\Gamma }}_q\left(z\right)$: $q$-gamma function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $d_{q}x$: $q$-differential, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, $q$: complex base, $z$: complex variable and $x$: real variable

Permalink:

http://dlmf.nist.gov/17.6.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

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17.6.29 $${}_2{}^{}\varphi _1^{}(\genfrac{}{}{0pt}{}{a,b}{c};q,z)=\left(\frac{-1}{2\pi \mathrm{i}}\right)\frac{{(a,b;q)}_{\mathrm{\infty }}}{{(q,c;q)}_{\mathrm{\infty }}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{{(q^{1+\zeta },c{q}^{\zeta };q)}_{\mathrm{\infty }}}{{(a{q}^{\zeta },b{q}^{\zeta };q)}_{\mathrm{\infty }}}\frac{\pi {(-z)}^{\zeta }}{\sin \left(\pi \zeta \right)}d\zeta ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${}_{r+1}{}^{}\varphi _s^{}(a_0,\mathrm{\dots },a_r;b_1,\mathrm{\dots },b_s;q,z)$ or ${}_{r+1}{}^{}\varphi _s^{}(\genfrac{}{}{0pt}{}{a_0,\mathrm{\dots },a_r}{b_1,\mathrm{\dots },b_s};q,z)$: basic hypergeometric (or $q$-hypergeometric) function, ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $\sin z$: sine function, $q$: complex base and $z$: complex variable

Permalink:

http://dlmf.nist.gov/17.6.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §17.6(v), §17.6 and Ch.17

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15: 19.19 Taylor and Related Series

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19.19.2 $$R_{-a}(𝐛;𝐳)=\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(𝐛,\mathrm{𝟏}-𝐳),$$ $c={\sum }_{j=1}^n{b}_j$, ,

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.19.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.3 $$R_{-a}(𝐛;𝐳)=z_n^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(c\right)}_N}{T}_N(b_1,\mathrm{\dots },b_{n-1};1-(z_1/z_n),\mathrm{\dots },1-(z_{n-1}/z_n)),$$ $c={\sum }_{j=1}^n{b}_j$, .

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer and $T_N(𝐛,𝐳)$: polynomial

Referenced by:

§19.19, §19.19, §19.28

Permalink:

http://dlmf.nist.gov/19.19.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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19.19.7 $$R_{-a}(\frac{\mathrm{𝟏}}{\mathrm{𝟐}};𝐳)=A^{-a}\sum _{N=0}^{\mathrm{\infty }}\frac{{\left(a\right)}_N}{{\left(\frac{1}{2}n\right)}_N}{T}_N(\frac{\mathrm{𝟏}}{\mathrm{𝟐}},𝐙),$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $n$: nonnegative integer, $N$: nonnegative integer, $T_N(𝐛,𝐳)$: polynomial, $𝐙$: variable and $A$

Referenced by:

§19.15, §19.36(i), §19.36(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.19.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.19 and Ch.19

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16: 5.18 $q$-Gamma and $q$-Beta Functions

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

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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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17: 18.27 $q$-Hahn Class

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

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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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18.27.19 $${\int }_0^{\mathrm{\infty }}\frac{S_n(x;q)S_m(x;q)}{{(-x,-q{x}^{-1};q)}_{\mathrm{\infty }}}dx=\frac{\ln \left(q^{-1}\right)}{q^n}\frac{{(q;q)}_{\mathrm{\infty }}}{{(q;q)}_n}{\delta }_{n,m},$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $dx$: differential of $x$ , $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), ${(a_1,a_2,\mathrm{\dots },a_r;q)}_n$: multiple $q$-Pochhammer symbol, $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E19

Encodings:

pMML, png, TeX

See also:

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

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18.27.20 $${\int }_0^{\mathrm{\infty }}{S}_n(q^{\frac{1}{2}}x;q)S_m(q^{\frac{1}{2}}x;q)\exp \left(-\frac{{(\ln x)}^2}{2\ln \left(q^{-1}\right)}\right)dx=\frac{\sqrt{2\pi q^{-1}\ln \left(q^{-1}\right)}}{q^n{(q;q)}_n}{\delta }_{n,m}.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $S_n(x;q)$: Stieltjes–Wigert polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $\ln z$: principal branch of logarithm function, ${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $q$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Permalink:

http://dlmf.nist.gov/18.27.E20

Encodings:

pMML, png, TeX

See also:

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

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18: 8.4 Special Values

… ►For $\mathrm{erf}\left(z\right)$, $\mathrm{erfc}\left(z\right)$, and $F\left(z\right)$, see §§7.2(i), 7.2(ii). For $E_n\left(z\right)$ see §8.19(i). … ►

8.4.4 $$\mathrm{\Gamma }(0,z)={\int }_z^{\mathrm{\infty }}{t}^{-1}{\mathrm{e}}^{-t}dt=E_1\left(z\right),$$

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Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral and $z$: complex variable

A&S Ref:

6.5.15

Permalink:

http://dlmf.nist.gov/8.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.13 $$\mathrm{\Gamma }(1-n,z)=z^{1-n}{E}_n\left(z\right),$$

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Symbols:

$E_p\left(z\right)$: generalized exponential integral, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

Referenced by:

§8.19(iv)

Permalink:

http://dlmf.nist.gov/8.4.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.14 $$Q(n+\frac{1}{2},z^2)=\mathrm{erfc}\left(z\right)+\frac{{\mathrm{e}}^{-z^2}}{\sqrt{\pi }}\sum _{k=1}^n\frac{z^{2k-1}}{{\left(\frac{1}{2}\right)}_k},$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Referenced by:

§8.4

Permalink:

http://dlmf.nist.gov/8.4.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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19: 18.17 Integrals

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18.17.36 $${\int }_{-1}^1{(1-x)}^{z-1}{(1+x)}^{\beta }{P}_n^{(\alpha ,\beta )}\left(x\right)dx=\frac{2^{\beta +z}\mathrm{\Gamma }\left(z\right)\mathrm{\Gamma }\left(1+\beta +n\right){\left(1+\alpha -z\right)}_n}{n!\mathrm{\Gamma }\left(1+\beta +z+n\right)},$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_n^{(\alpha ,\beta )}\left(x\right)$: Jacobi polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.38 $${\int }_0^1{P}_{2n}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(\frac{1}{2}-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>0$,

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.39 $${\int }_0^1{P}_{2n+1}\left(x\right)x^{z-1}dx=\frac{{(-1)}^n{\left(1-\frac{1}{2}z\right)}_n}{2{\left(\frac{1}{2}+\frac{1}{2}z\right)}_{n+1}},$$ $\mathrm{\Re }z>-1$.

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable, $n$: nonnegative integer and $x$: real variable

Keywords:

Mellin transform

Referenced by:

§18.17(vii)

Permalink:

http://dlmf.nist.gov/18.17.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §18.17(vii), §18.17(vii), §18.17 and Ch.18

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18.17.41_5 $${\int }_{-1}^1{C}_{\mathrm{\ell }}^{(\lambda )}\left(x\right)C_m^{(\lambda )}\left(x\right)C_n^{(\lambda )}\left(x\right){(1-x^2)}^{\lambda -\frac{1}{2}}dx=\frac{{\left(\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n}{\left(\lambda \right)}_{\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell }}{\left(\lambda \right)}_{\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m}{\left(2\lambda \right)}_{\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n}\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\right)\sqrt{\pi }}{(\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m-\frac{1}{2}n)!(\frac{1}{2}m+\frac{1}{2}n-\frac{1}{2}\mathrm{\ell })!(\frac{1}{2}n+\frac{1}{2}\mathrm{\ell }-\frac{1}{2}m)!\mathrm{\Gamma }\left(\lambda +\frac{1}{2}\mathrm{\ell }+\frac{1}{2}m+\frac{1}{2}n+1\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $C_n^{(\lambda )}\left(x\right)$: ultraspherical (or Gegenbauer) polynomial, $\mathrm{\ell }$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Andrews et al. (1999, Corollary 6.8.4); Straightforward from (18.18.22).(proved)

Referenced by:

§18.17(vii), Erratum (V1.2.0) Section 18.17

Permalink:

http://dlmf.nist.gov/18.17.E41_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.17(viii), §18.17(viii), §18.17 and Ch.18

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20: 19.20 Special Cases

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$R_N(𝐛;𝐳)={\displaystyle \frac{N!}{{\left(c\right)}_N}}{T}_N(𝐛,𝐳),$ $N=0,1,2,\mathrm{\dots }$,

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