Another Fourier-style expansion in series of Legendre functions (Q1571023)

In a previous paper [Proc. Lond. Math. Soc., III. Ser. 69, 629-672 (1994; Zbl 0807.42021)], the author established a Fourier-style expansion in \(-1< x< 1\) of the form \[ {1\over 2} [f(x+ 0)+ f(x- 0)]= a_0 P^\mu_\nu(x)+ \sum^\infty_{n=1} [a_n P^\mu_{\nu+ n}(x)+ a_{-n} P^\mu_{\nu- n}(x)], \] where the functions \(P^\mu_\nu(x)\) are the Legendre functions. The conditions for the validity of the above expansion require that \((1- t^2)^{-1/4}f(t)\in L(-1,1)\), \(f\) has bounded variation on a neighbourhood of a certain \(x\in(-1, 1)\), \(|\text{Re }\mu|< 1/2\), \(\nu\) is not half an integer, and \[ a_n= (-1)^n {\nu+ n+1/2\over 2\cos\nu\pi} \int^1_{- 1} f(t) P^{-\mu}_{\nu+ n}(-t) dt(x). \] The result established in the present paper is that by adding the condition that \(f\) has bounded variation on a neighbourhood of \(-x\) and \(c_n= (-1)^n 2a_n\) then \[ {1\over 2} [f(x+ 0)+ f(x- 0)]= c_0 P^\mu_\nu(x)+ \sum^\infty_{n=1} [c_{2n} P^\mu_{\nu+ 2n}(x)+ c_{-2n}P^\mu_{\nu- 2n}(x)]. \]