Novel summation formulas for Jacobi and Gegenbauer polynomials

Abstract

Thermal equilibrium of a two-dimensional model of mobile point-like charges with pair logarithmic interactions, immersed in a fixed neutralising background, is exactly solvable at a special coupling constant. Assuming gauge invariance of the statistical mechanics and the requirement of local charge neutrality in the bulk regime of the system, the exact solution implies explicit expressions for inverse elements of an infinite interaction matrix. This result is equivalent to new summation formulas over upper index for bilinear products of Jacobi and Gegenbauer polynomials which are complementary to standard completeness relations involving lower indices for these polynomials. We go beyond the physical input and derive multi-variable generalizations of the summation formulas.