Inner product and Gegenbauer polynomials in Sobolev space

In this paper we consider the system of functions G_(r,n)^α (x) (r∈N,n=0,1,…) which is orthogonal with respect to Sobolev-type inner product on (-1,1) and generated by Gegenbauer polynomials. The main goal work study some properties related {φ_(k,r) (x)}_(k≥0) {G_(r,n)^α (x)} functions. We conditions a function f(x) given in generalized for it be expandable into mixed Fourier series form f(x)~∑_(k=0)^(r-1)▒〖f^((k) ) (-1) (x+1)^k/k!+∑_(k=r)^∞▒〖G_(r,k)^α (f) 〗〗 φ_(r,k)^α (x), as well convergence series. second result proof recurrence formula (x)}_(k≥0). also discuss asymptotic these functions, represents latter our contribution.