Christoffel-Darboux formula for zonal spherical functions for the Gelfand pair \((U(n),U(n - 1))\) (Q2473984)

There is the well-known possibility of Fourier expansion of a function \(f(x)\) according to the orthonormal system of Legendre polynomials \(\{p_1\mid 1=0,1,2,\dots\}\), with the inner product \((f,g)= \int^1_{-1} f(x)\overline{g(x)}\,dx\) and the properties of this system: \[ f(x)= \sum^\infty_{l=0} (f,p_1)p_1(x), \] \[ \sum^k_{l=0} (f,p_1)p_1(x)= \int^1_{-1} f(x)\Biggl(\sum^k_{l=0} ((21+ 1)/2) P_1(x), P_1(y)\Biggr)\,dy, \] \[ \sum^k_{l=0} (21+ 1)P_1(x)P_1(y)= ((k+1)/(x- y) (P_{k+1}(x)P_k(y)- P_k(x) P_{k+1}(y)), \] (known as Christoffel-Darboux formula), \((1+2)P_{1+2}(x)-(21+3)\times P_{1+1}(x)+ (1+1)P_1(x)= 0\) (the recurrence relation for the \(P_1\mid 1=0,1,2,\dots\}\) system), where \(P_1(x)\) is the Legendre polynomial of degree 1. The author establishes similar properties for the zonal spherical functions for the Gelfand pair \((U(n),U(n-1))\), where \(U(n)\) is the unitary group of degree \(n\). Being closely related to the Jacobi polynomials, the zonal spherical functions are given by the orthogonal functions \(G_{p,q}\); \(p,q= 0,1,2,\dots\)\ . For this system there is known the following generating function \[ (1- 2\text{\,Re}(wz)+ |z|^2)^{1-n}= \sum^\infty_{p,q= 0} G_{p,q}(z) w^p\overline w^q,\quad (w,z\in C,|w|< 1,|z|< 1). \] The main results of the paper are: 1. for the system \(\{G_{p,q}\mid p,q= 0,1,2,\dots\}\) there exists a Christoffel-Darboux formula; 2. in some conditions, the Fourier expansion of \(f(x)\), defined on the unit, open disk \(|z|< 1\) in \(C\), \(f(x)= \sum^\infty_{p,q= 0} c_{p,q} G_{p,q}(z)\), converges pointwise.