Another Fourier-style expansion in series of Legendre functions

In Love (Proc. London Math. Soc. 69 (3) (1994) 629–672) I established, under suitable conditions, a Fourier-style expansion, in Legendre functions Pνμ(x) on the cut −1<x<1, of the form12{f(x+0)+f(x−0)}=a0Pνμ(x)+∑n=1∞{anPν+nμ(x)+a−nPν−nμ(x)}.In this paper I establish an expansion of similar form but in which the terms corresponding to odd values of n are absent. This is achieved by adding to the above an expansion of zero of similar form; most of this paper is devoted to establishing that expansion of zero. Expansions similar to these occur in Love and Hunter (Proc. London Math. Soc. (3) 64 (1992) 579–601), but under conditions which are more restrictive; the methods used in this paper are mostly quite different.