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21: 16.13 Appell Functions

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16.13.1 $$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$: first Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E1

Encodings:

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See also:

Annotations for §16.13 and Ch.16

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16.13.2 $$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$: second Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E2

Encodings:

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See also:

Annotations for §16.13 and Ch.16

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16.13.3 $$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_m{\left({\alpha }^{\prime }\right)}_n{\left(\beta \right)}_m{\left({\beta }^{\prime }\right)}_n}{{\left(\gamma \right)}_{m+n}m!n!}{x}^m{y}^n,$$ ,

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Defines:

$F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$: third Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E3

Encodings:

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See also:

Annotations for §16.13 and Ch.16

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16.13.4 $$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)=\sum _{m,n=0}^{\mathrm{\infty }}\frac{{\left(\alpha \right)}_{m+n}{\left(\beta \right)}_{m+n}}{{\left(\gamma \right)}_m{\left({\gamma }^{\prime }\right)}_{n}m!n!}{x}^m{y}^n,$$ .

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Defines:

$F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$: fourth Appell function

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial) and $!$: factorial (as in $n!$)

Permalink:

http://dlmf.nist.gov/16.13.E4

Encodings:

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Annotations for §16.13 and Ch.16

►Here and elsewhere it is assumed that neither of the bottom parameters $\gamma$ and ${\gamma }^{\prime }$ is a nonpositive integer. …

22: 5.6 Inequalities

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5.6.2 $$\frac{1}{\mathrm{\Gamma }\left(x\right)}+\frac{1}{\mathrm{\Gamma }\left(1/x\right)}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E2

Encodings:

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Annotations for §5.6(i), §5.6 and Ch.5

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5.6.3 $$\frac{1}{{(\mathrm{\Gamma }\left(x\right))}^2}+\frac{1}{{(\mathrm{\Gamma }\left(1/x\right))}^2}\le 2,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E3

Encodings:

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See also:

Annotations for §5.6(i), §5.6 and Ch.5

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5.6.4 .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $s$: real or complex variable and $x$: real variable

Referenced by:

§5.6(i)

Permalink:

http://dlmf.nist.gov/5.6.E4

Encodings:

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Annotations for §5.6(i), §5.6(i), §5.6 and Ch.5

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5.6.6 $$|\mathrm{\Gamma }\left(x+\mathrm{i}y\right)|\le |\mathrm{\Gamma }\left(x\right)|,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $x$: real variable and $y$: real variable

A&S Ref:

6.1.26

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E6

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Annotations for §5.6(ii), §5.6 and Ch.5

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5.6.8 $$\left|\frac{\mathrm{\Gamma }\left(z+a\right)}{\mathrm{\Gamma }\left(z+b\right)}\right|\le \frac{1}{{|z|}^{b-a}}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $z$: complex variable, $a$: real or complex variable and $b$: real or complex variable

Referenced by:

§5.6(ii)

Permalink:

http://dlmf.nist.gov/5.6.E8

Encodings:

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Annotations for §5.6(ii), §5.6 and Ch.5

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23: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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$\delta$	arbitrary small positive constant.
$\gamma$	Euler’s constant (§5.2(ii)).
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►The main functions treated in this chapter are the gamma function $\mathrm{\Gamma }\left(z\right)$, the psi function (or digamma function) $\psi \left(z\right)$, the beta function $\mathrm{B}(a,b)$, and the $q$-gamma function ${\mathrm{\Gamma }}_q\left(z\right)$. ►The notation $\mathrm{\Gamma }\left(z\right)$ is due to Legendre. Alternative notations for this function are: $\mathrm{\Pi }(z-1)$ (Gauss) and $(z-1)!$. …

24: 8.3 Graphics

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Figure 8.3.1: $\mathrm{\Gamma }(a,x)$, $a$ = 0. … Magnify

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Figure 8.3.2: $\gamma (a,x)$, $a$ = 0. … Magnify

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Figure 8.3.3: $\gamma (a,x)$, $a$ = 1, 2, 2. … Magnify

… ►Some monotonicity properties of ${\gamma }^{\ast }(a,x)$ and $\mathrm{\Gamma }(a,x)$ in the four quadrants of the ($a,x$)-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). … ►

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Figure 8.3.8: $\mathrm{\Gamma }(0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. …When $x=y=0$, $\mathrm{\Gamma }(0.25,0)=\mathrm{\Gamma }\left(0.25\right)=3.625\mathrm{\dots }$. Magnify 3D Help

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25: 16.1 Special Notation

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$p,q$	nonnegative integers.
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$\delta$	arbitrary small positive constant.
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►The main functions treated in this chapter are the generalized hypergeometric function ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$, the Appell (two-variable hypergeometric) functions $F_1(\alpha ;\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ;\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime };\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, and the Meijer $G$-function $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$. Alternative notations are ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$, and ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ for the generalized hypergeometric function, $F_1(\alpha ,\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_2(\alpha ,\beta ,{\beta }^{\prime };\gamma ,{\gamma }^{\prime };x,y)$, $F_3(\alpha ,{\alpha }^{\prime },\beta ,{\beta }^{\prime };\gamma ;x,y)$, $F_4(\alpha ,\beta ;\gamma ,{\gamma }^{\prime };x,y)$, for the Appell functions, and $G_{p,q}^{m,n}(z;𝐚;𝐛)$ for the Meijer $G$-function.

26: 18.25 Wilson Class: Definitions

… ►Table 18.25.1 lists the transformations of variable, orthogonality ranges, and parameter constraints that are needed in §18.2(i) for the Wilson polynomials $W_n(x;a,b,c,d)$, continuous dual Hahn polynomials $S_n(x;a,b,c)$, Racah polynomials $R_n(x;\alpha ,\beta ,\gamma ,\delta )$, and dual Hahn polynomials $R_n(x;\gamma ,\delta ,N)$. … ►

$$\gamma ,\delta >-1,\beta >N+\gamma .$$

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►The first four sets imply $\gamma +\delta >-2$, and the last four imply . …

27: 8.14 Integrals

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8.14.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\frac{\gamma (b,x)}{\mathrm{\Gamma }\left(b\right)}dx=\frac{{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }b>-1$,

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Symbols:

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

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E1

Encodings:

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Annotations for §8.14 and Ch.8

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8.14.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-ax}\mathrm{\Gamma }(b,x)dx=\mathrm{\Gamma }\left(b\right)\frac{1-{(1+a)}^{-b}}{a},$$ $\mathrm{\Re }a>-1$, $\mathrm{\Re }b>-1$.

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Symbols:

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

A&S Ref:

6.5.36

Referenced by:

§8.14, §8.14

Permalink:

http://dlmf.nist.gov/8.14.E2

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Annotations for §8.14 and Ch.8

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8.14.3 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\gamma (b,x)dx=-\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ , $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Referenced by:

§8.14

Permalink:

http://dlmf.nist.gov/8.14.E3

Encodings:

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See also:

Annotations for §8.14 and Ch.8

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8.14.4 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}\mathrm{\Gamma }(b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{a},$$ $\mathrm{\Re }a>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

A&S Ref:

6.5.37

Permalink:

http://dlmf.nist.gov/8.14.E4

Encodings:

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Annotations for §8.14 and Ch.8

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8.14.5 $${\int }_0^{\mathrm{\infty }}{x}^{a-1}{\mathrm{e}}^{-sx}\gamma (b,x)dx=\frac{\mathrm{\Gamma }\left(a+b\right)}{b{(1+s)}^{a+b}}F(1,a+b;1+b;1/(1+s)),$$ $\mathrm{\Re }s>0$, $\mathrm{\Re }\left(a+b\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.14.E5

Encodings:

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See also:

Annotations for §8.14 and Ch.8

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28: 30.16 Methods of Computation

… ►For $m=2$, $n=4$, ${\gamma }^2=10$, … ►If $|{\gamma }^2|$ is large, then we can use the asymptotic expansions referred to in §30.9 to approximate ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$. … ►If ${\lambda }_n^m\left({\gamma }^2\right)$ is known, then ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ can be found by summing (30.8.1). The coefficients $a_{n,r}^m({\gamma }^2)$ are computed as the recessive solution of (30.8.4) (§3.6), and normalized via (30.8.5). … ►The coefficients $a_{n,k}^m({\gamma }^2)$ calculated in §30.16(ii) can be used to compute $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$ from (30.11.3) as well as the connection coefficients $K_n^m(\gamma )$ from (30.11.10) and (30.11.11). …

29: 4.34 Derivatives and Differential Equations

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4.34.11 $$w=A\cosh \left(az\right)+B\sinh \left(az\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E11

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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4.34.12 $$w=(1/a)\sinh \left(az+c\right),$$

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Symbols:

$\sinh z$: hyperbolic sine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E12

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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4.34.13 $$w=(1/a)\cosh \left(az+c\right),$$

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Symbols:

$\cosh z$: hyperbolic cosine function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E13

Encodings:

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See also:

Annotations for §4.34 and Ch.4

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4.34.14 $$w=(1/a)\coth \left(az+c\right),$$

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Symbols:

$\coth z$: hyperbolic cotangent function, $a$: real or complex constant, $z$: complex variable, $w$: solution and $c$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.34.E14

Encodings:

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See also:

Annotations for §4.34 and Ch.4

►where $A,B,c$ are arbitrary constants. …

30: 8.13 Zeros

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§8.13(i) $x$-Zeros of ${\gamma }^{\ast }(a,x)$

►The function ${\gamma }^{\ast }(a,x)$ has no real zeros for $a\ge 0$. … ►

§8.13(ii) $\lambda$-Zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$

►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). ►

§8.13(iii) $a$-Zeros of ${\gamma }^{\ast }(a,x)$

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