3F2 functions of matrix argument

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21: 4.2 Definitions

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§4.2(iii) The Exponential Function

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§4.2(iv) Powers

… ►In particular, $z^0=1$, and if $a=n=1,2,3,\mathrm{\dots }$, then … …

22: 23.2 Definitions and Periodic Properties

… ►If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators. … ►

§23.2(ii) Weierstrass Elliptic Functions

… ►The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$, … ►For $j=1,2,3$, the function $\sigma \left(z\right)$ satisfies …More generally, if $j=1,2,3$, $k=1,2,3$, $j\ne k$, and $m,n\in \mathbb{Z}$, then …

23: 17.1 Special Notation

§17.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${}_r{}^{}\varphi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${}_r{}^{}\psi _s^{}(a_1,a_2,\mathrm{\dots },a_r;b_1,b_2,\mathrm{\dots },b_s;q,z)$, and the $q$-analogs of the Appell functions ${\mathrm{\Phi }}^{(1)}(a;b,b^{\prime };c;q;x,y)$, ${\mathrm{\Phi }}^{(2)}(a;b,b^{\prime };c,c^{\prime };q;x,y)$, ${\mathrm{\Phi }}^{(3)}(a,a^{\prime };b,b^{\prime };c;q;x,y)$, and ${\mathrm{\Phi }}^{(4)}(a,b;c,c^{\prime };q;x,y)$. ►Another function notation used is the “idem” function: …

24: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument $z$ and parameter $a$. … ►

Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(x) Modulus and Phase Functions

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25: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►The main functions treated in this chapter are the eigenvalues ${\lambda }_n^m\left({\gamma }^2\right)$ and the spheroidal wave functions ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$, and $S_n^{m(j)}(z,\gamma )$, $j=1,2,3,4$. …Meixner and Schäfke (1954) use $\mathrm{ps}$, $\mathrm{qs}$, $\mathrm{Ps}$, $\mathrm{Qs}$ for $\mathrm{𝖯𝗌}$, $\mathrm{𝖰𝗌}$, $\mathrm{𝑃𝑠}$, $\mathrm{𝑄𝑠}$, respectively. ►

Other Notations

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26: 1.10 Functions of a Complex Variable

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Phase (or Argument) Principle

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Analytic Functions

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§1.10(vi) Multivalued Functions

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§1.10(vii) Inverse Functions

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§1.10(xi) Generating Functions

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27: 25.11 Hurwitz Zeta Function

§25.11 Hurwitz Zeta Function

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§25.11(i) Definition

… ►The Riemann zeta function is a special case: … ►

§25.11(ii) Graphics

… ►where $h,k$ are integers with $1\le h\le k$ and $n=1,2,3,\mathrm{\dots }$. …

28: 11.10 Anger–Weber Functions

§11.10 Anger–Weber Functions

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§11.10(vi) Relations to Other Functions

… ► ►For $n=1,2,3,\mathrm{\dots }$, … ►

§11.10(viii) Expansions in Series of Products of Bessel Functions

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29: 25.1 Special Notation

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

$k,m,n$	nonnegative integers.
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primes	on function symbols: derivatives with respect to argument.

►The main function treated in this chapter is the Riemann zeta function $\zeta \left(s\right)$. … ►The main related functions are the Hurwitz zeta function $\zeta (s,a)$, the dilogarithm ${\mathrm{Li}}_2\left(z\right)$, the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ (also known as Jonquière’s function $\varphi (z,s)$), Lerch’s transcendent $\mathrm{\Phi }(z,s,a)$, and the Dirichlet $L$-functions $L(s,\chi )$.

30: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

… ►The $4m$ and $4m+1$ convergents are less than $I_x(a,b)$, and the $4m+2$ and $4m+3$ convergents are greater than $I_x(a,b)$. … ►

§8.17(vi) Sums

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