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13 Confluent Hypergeometric FunctionsNotation

§13.1 Special Notation

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

m integer.
n, s nonnegative integers.
x, y real variables.
z complex variable.
δ arbitrary small positive constant.
γ Euler’s constant (§5.2(ii)).
Γ(x) gamma function (§5.2(i)).
ψ(x) Γ(x)/Γ(x).

The main functions treated in this chapter are the Kummer functions M(a,b,z) and U(a,b,z), Olver’s function 𝐌(a,b,z), and the Whittaker functions Mκ,μ(z) and Wκ,μ(z).

Other notations are: F11(a;b;z)16.2(i)) and Φ(a;b;z) (Humbert (1920)) for M(a,b,z); Ψ(a;b;z) (Erdélyi et al. (1953a, §6.5)) for U(a,b,z); V(ba,b,z) (Olver (1997b, p. 256)) for ezU(a,b,z); Γ(1+2μ)κ,μ (Buchholz (1969, p. 12)) for Mκ,μ(z).

For an historical account of notations see Slater (1960, Chapter 1).