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11: 26.5 Lattice Paths: Catalan Numbers

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§26.5(i) Definitions

… ►(Sixty-six equivalent definitions of $C\left(n\right)$ are given in Stanley (1999, pp. 219–229).) …

12: 18.3 Definitions

§18.3 Definitions

… ►Table 18.3.1 provides the traditional definitions of Jacobi, Laguerre, and Hermite polynomials via orthogonality and standardization (§§18.2(i) and 18.2(iii)). … ►

Table 18.3.1: Orthogonality properties for classical OP’s: intervals, weight functions, standardizations, leading coefficients, and parameter constraints. …

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Name	$p_n(x)$	$(a,b)$	$w(x)$	$h_n$	$k_n$	${\tilde{k}}_n/k_n$	Constraints
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13: 4.1 Special Notation

… ►It is assumed the user is familiar with the definitions and properties of elementary functions of real arguments $x$. The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. …

14: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

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See also:

Annotations for §31.5 and Ch.31

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15: 4.28 Definitions and Periodicity

§4.28 Definitions and Periodicity

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16: 31.14 General Fuchsian Equation

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§31.14(i) Definitions

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17: 6.2 Definitions and Interrelations

§6.2 Definitions and Interrelations

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§6.2(i) Exponential and Logarithmic Integrals

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§6.2(ii) Sine and Cosine Integrals

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§6.2(iii) Auxiliary Functions

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18: 25.15 Dirichlet $L$-functions

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§25.15(i) Definitions and Basic Properties

►The notation $L(s,\chi )$ was introduced by Dirichlet (1837) for the meromorphic continuation of the function defined by the series ►

25.15.1 $$L(s,\chi )=\sum _{n=1}^{\mathrm{\infty }}\frac{\chi (n)}{n^s},$$ $\mathrm{\Re }s>1$,

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Defines:

$L(s,\chi )$: Dirichlet $L$-function

Symbols:

$\mathrm{\Re }$: real part, $n$: nonnegative integer, $s$: complex variable and $\chi (n)$: Dirichlet character

Keywords:

definition, infinite series, series representation

Source:

Apostol (1976, p. 249)

Referenced by:

(25.11.36), (25.11.36), §25.11(x), Erratum (V1.0.23) for Equation (25.11.36)

Permalink:

http://dlmf.nist.gov/25.15.E1

Encodings:

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See also:

Annotations for §25.15(i), §25.15 and Ch.25

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25.15.6 $$G(\chi )\equiv \sum _{r=1}^{k-1}\chi (r){\mathrm{e}}^{2\pi \mathrm{i}r/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer and $\chi (n)$: Dirichlet character

Keywords:

definition

Source:

Apostol (1976, p. 262)

Referenced by:

Erratum (V1.1.4) for Equation (25.15.6)

Permalink:

http://dlmf.nist.gov/25.15.E6

Encodings:

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Clarification (effective with 1.1.4):

The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi (k)=0$.

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.15(i), §25.15 and Ch.25

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19: 25.12 Polylogarithms

… ►The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): ►

25.12.1 $${\mathrm{Li}}_2\left(z\right)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2},$$ $|z|\le 1$.

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Defines:

${\mathrm{Li}}_2\left(z\right)$: dilogarithm

Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable

Keywords:

definition

Source:

Maximon (2003, (3.1), p. 2808)

A&S Ref:

27.7.2 (with $z=1-x$)

Referenced by:

§25.12(i), §25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E1

Encodings:

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See also:

Annotations for §25.12(i), §25.12 and Ch.25

… ►For real or complex $s$ and $z$ the polylogarithm ${\mathrm{Li}}_s\left(z\right)$ is defined by ►

25.12.10 $${\mathrm{Li}}_s\left(z\right)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^s}.$$

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Defines:

${\mathrm{Li}}_s\left(z\right)$: polylogarithm

Symbols:

$n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

definition, infinite series, series representation

Source:

Lewin (1981, p. 236)

Referenced by:

(25.14.3)

Permalink:

http://dlmf.nist.gov/25.12.E10

Encodings:

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See also:

Annotations for §25.12(ii), §25.12 and Ch.25

… ►The Fermi–Dirac and Bose–Einstein integrals are defined by …

20: 10.2 Definitions

§10.2 Definitions

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Bessel Functions of the Third Kind (Hankel Functions)

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Cylinder Functions

►The notation ${𝒞}_{\nu }\left(z\right)$ denotes $J_{\nu }\left(z\right)$, $Y_{\nu }\left(z\right)$, $H_{\nu }^{(1)}\left(z\right)$, $H_{\nu }^{(2)}\left(z\right)$, or any nontrivial linear combination of these functions, the coefficients in which are independent of $z$ and $\nu$. …