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31 Heun FunctionsNotation

§31.1 Special Notation

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

x, y real variables.
z, ζ, w, W complex variables.
j, k, , m, n nonnegative integers.
a complex parameter, |a|1,a1.
q,α,β,γ,δ,ϵ,ν complex parameters.

The main functions treated in this chapter are H(a,q;α,β,γ,δ;z), (s1,s2)𝐻𝑓m(a,qm;α,β,γ,δ;z), (s1,s2)𝐻𝑓mν(a,qm;α,β,γ,δ;z), and the polynomial 𝐻𝑝n,m(a,qn,m;n,β,γ,δ;z). These notations were introduced by Arscott in Ronveaux (1995, pp. 34–44). Sometimes the parameters are suppressed.