Pochhammer integral

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21: 10.22 Integrals

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10.22.36 $$\frac{1}{\mathrm{\Gamma }\left(\alpha \right)}{\int }_0^x{(x-t)}^{\alpha -1}{J}_{\nu }\left(t\right)dt=2^{\alpha }\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_k}{k!}{J}_{\nu +\alpha +2k}\left(x\right),$$ $\mathrm{\Re }\alpha >0,\mathrm{\Re }\nu \ge 0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $x$: real variable, $\nu$: complex parameter and $\alpha$: positive integer

A&S Ref:

11.2.3 and 11.2.4

Referenced by:

§10.22(ii), §10.22(ii)

Permalink:

http://dlmf.nist.gov/10.22.E36

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The Pochhammer symbol now links to its definition.

See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

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22: 2.11 Remainder Terms; Stokes Phenomenon

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2.11.6 $$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}$$

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm and $E_p\left(z\right)$: generalized exponential integral

Referenced by:

§2.11(ii), §2.11(iii), §2.11(iii), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.7 $$E_p\left(z\right)\sim \frac{2\pi \mathrm{i}{\mathrm{e}}^{-p\pi \mathrm{i}}}{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{\mathrm{\infty }}{(-1)}^s\frac{{\left(p\right)}_s}{z^s},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $\mathrm{i}$: imaginary unit

Referenced by:

§2.11(ii), §2.11(iv), §2.11(iv), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(ii), §2.11 and Ch.2

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2.11.10 $$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\sum _{s=0}^{n-1}{(-1)}^s\frac{{\left(p\right)}_s}{z^s}+{(-1)}^n\frac{2\pi }{\mathrm{\Gamma }\left(p\right)}{z}^{p-1}{F}_{n+p}\left(z\right),$$

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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, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral and $F_p\left(z\right)$: terminant function

Referenced by:

§2.11(iii), §2.11(iv)

Permalink:

http://dlmf.nist.gov/2.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §2.11(iii), §2.11 and Ch.2

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23: 31.10 Integral Equations and Representations

§31.10 Integral Equations and Representations

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Kernel Functions

… ►where $\mathrm{\Re }\gamma >0$, $\mathrm{\Re }\delta >0$, and $C$ be the Pochhammer double-loop contour about 0 and 1 (as in §31.9(i)). … ►

Kernel Functions

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24: 18.30 Associated OP’s

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18.30.10 $${\int }_0^{\mathrm{\infty }}{L}_n^{\lambda }(x;c)L_m^{\lambda }(x;c)w^{\lambda }(x,c)dx=\frac{\mathrm{\Gamma }\left(n+c+\lambda +1\right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ $c+\lambda >-1$, $c\ge 0$, or $c+\lambda \ge 0$, $c>-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $L_n^{\lambda }(x;c)$: associated Laguerre polynomial, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Referenced by:

§18.30(iii)

Permalink:

http://dlmf.nist.gov/18.30.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(iii), §18.30 and Ch.18

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18.30.17 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{𝒫}_n^{\lambda }(x;\varphi ,c){𝒫}_m^{\lambda }(x;\varphi ,c)w^{(\lambda )}(x,\varphi ,c)dx=\frac{\mathrm{\Gamma }\left(n+c+2\lambda \right)\mathrm{\Gamma }\left(c+1\right)}{{\left(c+1\right)}_n}{\delta }_{n,m},$$ or ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${𝒫}_n^{\lambda }(x;\varphi ,c)$: associated Meixner–Pollaczek polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Sources:

Askey and Wimp (1984, (1.9)); Pollaczek (1950, formula following (16))

Permalink:

http://dlmf.nist.gov/18.30.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §18.30(v), §18.30 and Ch.18

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25: 18.34 Bessel Polynomials

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18.34.5_5 $$\frac{2^{1-a}}{\mathrm{\Gamma }\left(1-a\right)}{\int }_0^{\mathrm{\infty }}{y}_n(x;a)y_m(x;a)x^{a-2}{\mathrm{e}}^{-2x^{-1}}dx=\frac{1-a}{1-a-2n}\frac{n!}{{\left(2-a-n\right)}_n}{\delta }_{n,m},$$ $m,n=0,1,\mathrm{\dots },N=\lceil -(1+a)/2\rceil$.

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Symbols:

$y_n(x;a)$: Bessel polynomial, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\lceil x\rceil$: ceiling of $x$ , $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $N$: positive integer, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Lesky (1996, p.183, case a.3))(proved)

Sources:

Romanovski (1929, Type V); Koekoek et al. (2010, (9.13.2))

Referenced by:

(18.34.7_3), §18.34(ii), §18.34(iii), Erratum (V1.2.0) Section 18.34

Permalink:

http://dlmf.nist.gov/18.34.E5_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.34(ii), §18.34 and Ch.18

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26: 15.6 Integral Representations

§15.6 Integral Representations

►The function $𝐅(a,b;c;z)$ (not $F(a,b;c;z)$) has the following integral representations: … ►Note that (15.6.8) can be rewritten as a fractional integral. … ►

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Figure 15.6.1: $t$-plane. … Magnify

27: 18.35 Pollaczek Polynomials

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18.35.6_5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b,c)P_m^{(\lambda )}(x;a,b,c)w^{(\lambda )}(x;a,b,c)dx=\frac{\mathrm{\Gamma }\left(c+1\right)\mathrm{\Gamma }\left(2\lambda +c+n\right)}{{\left(c+1\right)}_n(\lambda +a+c+n)}{\delta }_{n,m},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Pollaczek (1950, (3))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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28: 8.7 Series Expansions

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8.7.2 $$\gamma (a,x+y)-\gamma (a,x)=\mathrm{\Gamma }(a,x)-\mathrm{\Gamma }(a,x+y)={\mathrm{e}}^{-x}{x}^{a-1}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{{(-x)}^n}(1-{\mathrm{e}}^{-y}{e}_n(y)),$$ .

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.30

Permalink:

http://dlmf.nist.gov/8.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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8.7.5 $${\gamma }^{\ast }(a,z)={\mathrm{e}}^{-\frac{1}{2}z}\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(1-a\right)}_n}{\mathrm{\Gamma }\left(n+a+1\right)}\left(2n+1\right){𝗂}_n^{(1)}\left(\frac{1}{2}z\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, ${𝗂}_n^{(1)}\left(z\right)$: modified spherical Bessel function, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Referenced by:

§8.7

Permalink:

http://dlmf.nist.gov/8.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.7 and Ch.8

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8.7.6 $$\mathrm{\Gamma }(a,x)=x^a{\mathrm{e}}^{-x}\sum _{n=0}^{\mathrm{\infty }}\frac{L_n^{(a)}\left(x\right)}{n+1},$$ $x>0$, .

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Symbols:

$L_n^{(\alpha )}\left(x\right)$: Laguerre (or generalized Laguerre) polynomial, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable, $a$: parameter and $n$: nonnegative integer

Proof sketch:

Integrate (18.12.13) from $z=0$ to $z=1$ and for the integral on the left-hand side use the substitution $z=\frac{t}{1+t}$. The resulting integral is (8.6.5). The constraint follows from (18.15.14).

Referenced by:

Erratum (V1.1.8) for Equation (8.7.6)

Permalink:

http://dlmf.nist.gov/8.7.E6

Encodings:

pMML, png, TeX

Addition (effective with 1.1.8):

The constraint was updated to include .

Suggested 2022-10-14 by Walter Gautschi

See also:

Annotations for §8.7 and Ch.8

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29: Errata

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18.28.8 $$\frac{1}{2\pi }{\int }_0^{\pi }{Q}_n(\cos \theta ;a,b|q)Q_m(\cos \theta ;a,b|q){\left|\frac{{({\mathrm{e}}^{2i\theta };q)}_{\mathrm{\infty }}}{{(a{\mathrm{e}}^{\mathrm{i}\theta },b{\mathrm{e}}^{\mathrm{i}\theta };q)}_{\mathrm{\infty }}}\right|}^{2}d\theta =\frac{{\delta }_{n,m}}{{(q^{n+1},ab{q}^n;q)}_{\mathrm{\infty }}},$$ $a,b\in ℝ$ or $a=\overline{b}$; $ab\ne 1$; $|a|,|b|\le 1$

The constraint which originally stated that “” has been updated to be “$ab\ne 1$”.

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17.13.3 $${\int }_0^{\mathrm{\infty }}{t}^{\alpha -1}\frac{{(-t{q}^{\alpha +\beta };q)}_{\mathrm{\infty }}}{{(-t;q)}_{\mathrm{\infty }}}dt=\frac{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\beta \right)}{{\mathrm{\Gamma }}_q\left(1-\alpha \right){\mathrm{\Gamma }}_q\left(\alpha +\beta \right)}$$

Originally the differential was identified incorrectly as $d_{q}t$; the correct differential is $dt$.

Reported 2011-04-08.

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30: 10.59 Integrals

§10.59 Integrals

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10.59.1

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Symbols:

$P_n\left(x\right)$: Legendre polynomial, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝗃}_n\left(z\right)$: spherical Bessel function of the first kind and $n$: integer

A&S Ref:

11.4.26

Referenced by:

§10.59

Permalink:

http://dlmf.nist.gov/10.59.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.59 and Ch.10

… ►For an integral representation of the Dirac delta in terms of a product of spherical Bessel functions of the first kind see §1.17(ii), and for a generalization see Maximon (1991). ►Additional integrals can be obtained by combining the definitions (10.47.3)–(10.47.9) with the results given in §10.22 and §10.43. For integrals of products see also Mehrem et al. (1991).