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1: 15.2 Definitions and Analytical Properties

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§15.2(i) Gauss Series

►The hypergeometric function $F(a,b;c;z)$ is defined by the Gauss series … ►The principal branch of $𝐅(a,b;c;z)$ is an entire function of $a$, $b$, and $c$. …The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. … ►For example, when $a=-m$, $m=0,1,2,\mathrm{\dots }$, and $c\ne 0,-1,-2,\mathrm{\dots }$, $F(a,b;c;z)$ is a polynomial: …

2: 23.15 Definitions

§23.15 Definitions

… ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, …If, as a function of $q$, $f(\tau )$ is analytic at $q=0$, then $f(\tau )$ is called a modular form. … ►

Dedekind’s Eta Function (or Dedekind Modular Function)

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3: 16.13 Appell Functions

§16.13 Appell Functions

►The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${}_2{}^{}F_1^{}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): … ►

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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4: 14.19 Toroidal (or Ring) Functions

§14.19 Toroidal (or Ring) Functions

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§14.19(i) Introduction

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§14.19(ii) Hypergeometric Representations

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§14.19(iv) Sums

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§14.19(v) Whipple’s Formula for Toroidal Functions

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5: 5.15 Polygamma Functions

§5.15 Polygamma Functions

►The functions ${\psi }^{(n)}\left(z\right)$, $n=1,2,\mathrm{\dots }$, are called the polygamma functions. In particular, ${\psi }^{\prime }\left(z\right)$ is the trigamma function; ${\psi }^{\prime \prime }$, ${\psi }^{(3)}$, ${\psi }^{(4)}$ are the tetra-, penta-, and hexagamma functions respectively. Most properties of these functions follow straightforwardly by differentiation of properties of the psi function. … ►For $B_{2k}$ see §24.2(i). …

6: 9.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►

$k$	nonnegative integer, except in §9.9(iii).
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►The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). ►Other notations that have been used are as follows: $\mathrm{Ai}\left(-x\right)$ and $\mathrm{Bi}\left(-x\right)$ for $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ (Jeffreys (1928), later changed to $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$); $U(x)=\sqrt{\pi }\mathrm{Bi}\left(x\right)$, $V(x)=\sqrt{\pi }\mathrm{Ai}\left(x\right)$ (Fock (1945)); $A(x)=3^{-1/3}\pi \mathrm{Ai}\left(-3^{-1/3}x\right)$ (Szegő (1967, §1.81)); $e_0(x)=\pi \mathrm{Hi}(-x)$, ${\tilde{e}}_0(x)=-\pi \mathrm{Gi}(-x)$ (Tumarkin (1959)).

7: 31.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) ► ►►

$x$, $y$	real variables.
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►The main functions treated in this chapter are $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, $(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$, and the polynomial ${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$. …Sometimes the parameters are suppressed.

8: 11.9 Lommel Functions

§11.9 Lommel Functions

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§11.9(ii) Expansions in Series of Bessel Functions

… ►For further information on Lommel functions see Watson (1944, §§10.7–10.75) and Babister (1967, Chapter 3). … ►

9: 9.12 Scorer Functions

§9.12 Scorer Functions

… ►where … ►

§9.12(ii) Graphs

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Functions and Derivatives

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10: 5.2 Definitions

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§5.2(i) Gamma and Psi Functions

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Euler’s Integral

… ►It is a meromorphic function with no zeros, and with simple poles of residue ${(-1)}^n/n!$ at $z=-n$. … ►

5.2.2 $$\psi \left(z\right)={\mathrm{\Gamma }}^{\prime }\left(z\right)/\mathrm{\Gamma }\left(z\right),$$ $z\ne 0,-1,-2,\mathrm{\dots }$.

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Defines:

$\psi \left(z\right)$: psi (or digamma) function

Symbols:

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

A&S Ref:

6.3.1

Permalink:

http://dlmf.nist.gov/5.2.E2

Encodings:

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See also:

Annotations for §5.2(i), §5.2(i), §5.2 and Ch.5

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5.2.3 $$\gamma =\underset{n\to \mathrm{\infty }}{lim}\left(1+\frac{1}{2}+\frac{1}{3}+\mathrm{\cdots }+\frac{1}{n}-\ln n\right)=\mathrm{0.57721\hspace{0.33em}56649\hspace{0.33em}01532\hspace{0.33em}86060}\mathrm{\dots }.$$

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Defines:

$\gamma$: Euler’s constant

Symbols:

$\ln z$: principal branch of logarithm function and $n$: nonnegative integer

A&S Ref:

6.1.3 (where the 10D value is given, and Table 1.1 where the 24D value is given.)

Notes:

For more digits see OEIS Sequence A001620; see also Sloane (2003).

Referenced by:

§4.4(iii)

Permalink:

http://dlmf.nist.gov/5.2.E3

Encodings:

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See also:

Annotations for §5.2(ii), §5.2 and Ch.5

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