generalized hypergeometric function 0F2

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21: 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). …

22: 5.12 Beta Function

§5.12 Beta Function

… ►In (5.12.1)–(5.12.4) it is assumed $\mathrm{\Re }a>0$ and $\mathrm{\Re }b>0$. ►

Euler’s Beta Integral

… ►In (5.12.8) the fractional powers have their principal values when $w>0$ and $z>0$, and are continued via continuity. … ►

Pochhammer’s Integral

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23: 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.

24: 19.2 Definitions

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§19.2(i) General Elliptic Integrals

►Let $s^2(t)$ be a cubic or quartic polynomial in $t$ with simple zeros, and let $r(s,t)$ be a rational function of $s$ and $t$ containing at least one odd power of $s$. … ►

§19.2(iv) A Related Function: $R_C(x,y)$

… ►where the Cauchy principal value is taken if . … ►In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). …

25: 4.2 Definitions

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§4.2(i) The Logarithm

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§4.2(ii) Logarithms to a General Base $a$

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§4.2(iii) The Exponential Function

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§4.2(iv) Powers

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Powers with General Bases

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26: 10.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►For the spherical Bessel functions and modified spherical Bessel functions the order $n$ is a nonnegative integer. For the other functions when the order $\nu$ is replaced by $n$, it can be any integer. For the Kelvin functions the order $\nu$ is always assumed to be real. … ►For older notations see British Association for the Advancement of Science (1937, pp. xix–xx) and Watson (1944, Chapters 1–3).

27: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►In other cases the general theory of (12.2.2) is available. … ►

§12.14(ii) Values at $z=0$ and Wronskian

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

Confluent Hypergeometric Functions

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28: 4.37 Inverse Hyperbolic Functions

§4.37 Inverse Hyperbolic Functions

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§4.37(i) General Definitions

►The general values of the inverse hyperbolic functions are defined by … ►

Other Inverse Functions

… ►With $k\in \mathbb{Z}$, the general solutions of the equations …

29: 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. ►The generators of a given lattice $𝕃$ are not unique. …In general, if … ►

§23.2(ii) Weierstrass Elliptic Functions

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30: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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§4.23(i) General Definitions

►The general values of the inverse trigonometric functions are defined by … ►Care needs to be taken on the cuts, for example, if then $1/(x+\mathrm{i}0)=(1/x)-\mathrm{i}0$. … ►With $k\in \mathbb{Z}$, the general solutions of the equations …