Hemicylindrical Function

A function $S_n(z)$ which satisfies the Recurrence Relation

$$S_{n-1}(z)-S_{n+1}(z)=2S_n'(z)$$

together with

$$S_1(z)=-S_0'(z)$$

is called a hemicylindrical function.

References

Sonine, N. ``Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries.'' Math. Ann. 16, 1-9 and 71-80, 1880.

Watson, G. N. ``Hemi-Cylindrical Functions.'' §10.8 in A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge, England: Cambridge University Press, p. 353, 1966.