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11: 9.1 Special Notation

… ►Other notations that have been used are as follows:

\operatorname{Ai}\left(-x\right)

and

\operatorname{Bi}\left(-x\right)

for

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

(Jeffreys (1928), later changed to

\operatorname{Ai}\left(x\right)

and

\operatorname{Bi}\left(x\right)

);

U(x)=\sqrt{\pi}\operatorname{Bi}\left(x\right)

V(x)=\sqrt{\pi}\operatorname{Ai}\left(x\right)

(Fock (1945));

A(x)=3^{-\ifrac{1}{3}}\pi\operatorname{Ai}\left(-3^{-\ifrac{1}{3}}x\right)

(Szegő (1967, §1.81));

e_{0}(x)=\pi\operatorname{Hi}(-x)

\widetilde{e}_{0}(x)=-\pi\operatorname{Gi}(-x)

(Tumarkin (1959)).

12: 26.1 Special Notation

… ►Many combinatorics references use the rising and falling factorials: … ►Other notations for

s\left(n,k\right)

, the Stirling numbers of the first kind, include

S_{n}^{(k)}

(Abramowitz and Stegun (1964, Chapter 24), Fort (1948)),

S_{n}^{k}

(Jordan (1939), Moser and Wyman (1958a)),

\genfrac{(}{)}{0.0pt}{}{n-1}{k-1}B_{n-k}^{(n)}

(Milne-Thomson (1933)),

(-1)^{n-k}S_{1}(n-1,n-k)

(Carlitz (1960), Gould (1960)),

(-1)^{n-k}\left[n\atop k\right]

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)). ►Other notations for

S\left(n,k\right)

, the Stirling numbers of the second kind, include

\mathscr{S}^{(k)}_{n}

(Fort (1948)),

\mathfrak{S}_{n}^{k}

(Jordan (1939)),

\sigma_{n}^{k}

(Moser and Wyman (1958b)),

\genfrac{(}{)}{0.0pt}{}{n}{k}B_{n-k}^{(-k)}

(Milne-Thomson (1933)),

S_{2}(k,n-k)

(Carlitz (1960), Gould (1960)),

\left\{n\atop k\right\}

(Knuth (1992), Graham et al. (1994), Rosen et al. (2000)), and also an unconventional symbol in Abramowitz and Stegun (1964, Chapter 24).

13: 8 Incomplete Gamma and Related
Functions

…

14: 12.18 Methods of Computation

… ►These include the use of power-series expansions, recursion, integral representations, differential equations, asymptotic expansions, and expansions in series of Bessel functions. …

15: 12 Parabolic Cylinder Functions

…

16: 17 q-Hypergeometric and Related Functions

…

17: 28 Mathieu Functions and Hill’s Equation

…

18: 29 Lamé Functions

…

19: 30 Spheroidal Wave Functions

…

20: 33 Coulomb Functions

…