double gamma function

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11: 13.4 Integral Representations

… ► … ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. … ►where the contour of integration separates the poles of $\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(1+a-b+t\right)$ from those of $\mathrm{\Gamma }\left(-t\right)$. …where the contour of integration passes all the poles of $\mathrm{\Gamma }\left(b-1+t\right)\mathrm{\Gamma }\left(t\right)$ on the right-hand side.

12: 31.9 Orthogonality

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31.9.2 $${\int }_{\zeta }^{(1+,0+,1-,0-)}{t}^{\gamma -1}{(1-t)}^{\delta -1}{(t-a)}^{ϵ-1}{w}_m(t)w_k(t)dt={\delta }_{m,k}{\theta }_m.$$

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Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\zeta$: point and ${\theta }_m$: normalization constant

Permalink:

http://dlmf.nist.gov/31.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.9(i), §31.9 and Ch.31

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13: 33.5 Limiting Forms for Small $\rho$, Small $|\eta |$, or Large $\mathrm{\ell }$

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§33.5(i) Small $\rho$

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§33.5(ii) $\eta =0$

… ►For the functions $𝗃$, $𝗒$, $J$, $Y$ see §§10.47(ii), 10.2(ii). … ►where $\gamma$ is Euler’s constant (§5.2(ii)). ►

§33.5(iv) Large $\mathrm{\ell }$

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14: 16.14 Partial Differential Equations

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§16.14(i) Appell Functions

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§16.14(ii) Other Functions

►In addition to the four Appell functions there are $24$ other sums of double series that cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, and which satisfy pairs of linear partial differential equations of the second order. … ►

16.14.5 $$G_2(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\alpha +m\right)\mathrm{\Gamma }\left({\alpha }^{\prime }+n\right)\mathrm{\Gamma }\left(\beta +n-m\right)\mathrm{\Gamma }\left({\beta }^{\prime }+m-n\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left({\alpha }^{\prime }\right)\mathrm{\Gamma }\left(\beta \right)\mathrm{\Gamma }\left({\beta }^{\prime }\right)}\frac{x^m{y}^n}{m!n!},$$ , ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.14(ii), §16.14 and Ch.16

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16.14.6 $$G_3(\alpha ,{\alpha }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\alpha +2n-m\right)\mathrm{\Gamma }\left({\alpha }^{\prime }+2m-n\right)}{\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }\left({\alpha }^{\prime }\right)}\frac{x^m{y}^n}{m!n!},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.14(ii), §16.14 and Ch.16

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15: Bibliography C

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Determination of $\nu$-zeros of Hankel functions. Comput. Phys. Comm. 32 (3), pp. 333–339.

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Notes:

Double-precision Fortran, maximum accuracy 7S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

1st item.

Encodings:

BibTeX

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On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

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Algorithm 597: Sequence of modified Bessel functions of the first kind. ACM Trans. Math. Software 9 (2), pp. 242–245.

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Notes:

Double-precision Fortran, maximum accuracy 24S.

External Links:

ISSN 0098-3500, Document, GAMS (module 597; package TOMS), ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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Algorithm 715: SPECFUN – A portable FORTRAN package of special function routines and test drivers. ACM Trans. Math. Software 19 (1), pp. 22–32.

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Notes:

Double precision, maximum accuracy 18S.

External Links:

ISSN 0098-3500, Document, Remark. GAMS (module 715; package TOMS), ZentralBlatt

Referenced by:

3rd item, 6th item, 4th item, 2nd item, 2nd item, 2nd item, 2nd item.

Encodings:

BibTeX

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A Fortran subroutine for the Bessel function $J_n(x)$ of order $0$ to $10$ . Comput. Phys. Comm. 21 (1), pp. 109–118.

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Notes:

Double-precision Fortran, maximum accuracy 16S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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16: 31.10 Integral Equations and Representations

… ►The weight function is given by … ►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)). Then the integral equation (31.10.1) is satisfied by $w(z)=w_m(z)$ and $W(z)={\kappa }_m{w}_m(z)$, where $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ and ${\kappa }_m$ is the corresponding eigenvalue. … ►Fuchs–Frobenius solutions $W_m(z)={\tilde{\kappa }}_m{z}^{-\alpha }H\mathrm{\ell }(1/a,q_m;\alpha ,\alpha -\gamma +1,\alpha -\beta +1,\delta ;1/z)$ are represented in terms of Heun functions $w_m(z)=(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$ by (31.10.1) with $W(z)=W_m(z)$, $w(z)=w_m(z)$, and with kernel chosen from … ►The weight function is …

17: Bibliography S

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Coulomb functions analytic in the energy. Comput. Phys. Comm. 25 (1), pp. 87–95.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

§33.1, §33.14(iv), 9th item.

Encodings:

BibTeX

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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.

External Links:

ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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Algorithm AS 239. Chi-squared and incomplete gamma integral. Appl. Statist. 37 (3), pp. 466–473.

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Notes:

Double-precision Fortran.

External Links:

FullText, Code

Referenced by:

5th item.

Encodings:

BibTeX

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Calculation of the gamma function by Stirling’s formula. Math. Comp. 25 (114), pp. 317–322.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§5.11(i), §5.11(ii).

Encodings:

BibTeX

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Automatic computing methods for special functions. II. The exponential integral $E_n(x)$ . J. Res. Nat. Bur. Standards Sect. B 78B, pp. 199–216.

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Notes:

Double-precision Fortran, maximum accuracy 16S.

External Links:

MathReview (Bernard H. Rosman), ZentralBlatt

Referenced by:

§8.25(v), §8.28(vi).

Encodings:

BibTeX

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18: Guide to Searching the DLMF

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any double-quoted sequence of textual words and numbers.

… ►DLMF search is generally case-insensitive except when it is important to be case-sensitive, as when two different special functions have the same standard names but one name has a lower-case initial and the other is has an upper-case initial, such as si and Si, gamma and Gamma. … ►

All the Greek Letters (gamma vs. Gamma, sigma vs. Sigma, etc.)

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All the inverse trigonometric functions (arcsin vs. Arcsin, etc.).

… ►For example, you may want equations that contain trigonometric functions, but you don’t care which trigonometric function. …

19: Bibliography T

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Parabolic cylinder functions $U(n,x)$ for natural $n$ and positive $x$ . Comput. Phys. Commun. 69, pp. 415–419.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 21D.

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

Encodings:

BibTeX

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Asymptotic inversion of incomplete gamma functions. Math. Comp. 58 (198), pp. 755–764.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§8.12, §8.12.

Encodings:

BibTeX

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item, §33.23(vi), 3rd item.

Encodings:

BibTeX

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20: Bibliography B

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A program for computing the Fermi-Dirac functions. Comput. Phys. Comm. 21 (3), pp. 315–322.

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Notes:

Double-precision Fortran, maximum accuracy 10D.

External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

External Links:

Document, Code

Referenced by:

1st item, 4th item.

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. Quart. J. Math., Oxford Ser. 11, pp. 249–270.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§15.16, (16.15.4), §16.15, §16.16(i).

Encodings:

BibTeX

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Expansions of Appell’s double hypergeometric functions. II. Quart. J. Math., Oxford Ser. 12, pp. 112–128.

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External Links:

Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§16.16(i).

Encodings:

BibTeX

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