orthogonal functions with respect to weighted summation

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21: 30.1 Special Notation

… ►(For other notation see Notation for the Special Functions.) … ►These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). 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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22: 16.2 Definition and Analytic Properties

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§16.2(i) Generalized Hypergeometric Series

… ► ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … ►

Polynomials

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§16.2(v) Behavior with Respect to Parameters

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23: 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

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§8.17(vi) Sums

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§8.17(vii) Addendum to 8.17(i) Definitions and Basic Properties

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

§12.14 The Function $W(a,x)$

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§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

… ►The expansions for the derivatives corresponding to (12.14.25), (12.14.26), and (12.14.31) may be obtained by formal term-by-term differentiation with respect to $t$; compare the analogous results in §§12.10(ii)–12.10(v). …

25: 4.23 Inverse Trigonometric Functions

§4.23 Inverse Trigonometric Functions

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Other Inverse Functions

… ►are respectively … ►

§4.23(viii) Gudermannian Function

… ►The inverse Gudermannian function is given by …

26: 1.10 Functions of a Complex Variable

… ►Analytic continuation is a powerful aid in establishing transformations or functional equations for complex variables, because it enables the problem to be reduced to: (a) deriving the transformation (or functional equation) with real variables; followed by (b) finding the domain on which the transformed function is analytic. … ►

§1.10(vi) Multivalued Functions

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§1.10(vii) Inverse Functions

… ►For each $t\in [a,b)$, $f(z,t)$ is analytic in $D$; $f(z,t)$ is a continuous function of both variables when $z\in D$ and $t\in [a,b)$; the integral (1.10.18) converges at $b$, and this convergence is uniform with respect to $z$ in every compact subset $S$ of $D$. … ►

§1.10(xi) Generating Functions

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27: 23.2 Definitions and Periodic Properties

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§23.2(i) Lattices

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§23.2(ii) Weierstrass Elliptic Functions

… ►When it is important to display the lattice with the functions they are denoted by $\mathrm{\wp }\left(z|𝕃\right)$, $\zeta \left(z|𝕃\right)$, and $\sigma \left(z|𝕃\right)$, respectively. …

28: 17.1 Special Notation

§17.1 Special Notation

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

$k,j,m,n,r,s$	nonnegative integers.
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… ► ►Another function notation used is the “idem” function: …

29: 14.1 Special Notation

§14.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►Multivalued functions take their principal values (§4.2(i)) unless indicated otherwise. … ► ►Among other notations commonly used in the literature Erdélyi et al. (1953a) and Olver (1997b) denote ${𝖯}_{\nu }^{\mu }\left(x\right)$ and ${𝖰}_{\nu }^{\mu }\left(x\right)$ by ${\mathrm{P}}_{\nu }^{\mu }(x)$ and ${\mathrm{Q}}_{\nu }^{\mu }(x)$, respectively. …

30: 35.1 Special Notation

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

$a,b$	complex variables.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu }\left(𝐓\right)$ and (of the second kind) $B_{\nu }\left(𝐓\right)$; confluent hypergeometric (of the first kind) ${}_1{}^{}F_1^{}(a;b;𝐓)$ or ${}_1{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};𝐓)$ and (of the second kind) $\mathrm{\Psi }(a;b;𝐓)$; Gaussian hypergeometric ${}_2{}^{}F_1^{}(a_1,a_2;b;𝐓)$ or ${}_2{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};𝐓)$; generalized hypergeometric ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).