Product Formulas and Nicholson-Type Integrals for Jacobi Functions. I: Summary of Results

Nicholson’s formula gives a generalization of the relation sin ² x + cos ² x = 1 to the case of Bessel functions. We present a similar result which relates the sum of squares of the Jacobi functions Pₙ(α ,β ) (x) and Qₙ(α ,β ) (x) to an integral over a single Jacobi function of the second kind, with the integrand positive. The Nicholson-type formula is a special case of a general product formula for two Jacobi functions of the second kind with different arguments, Qₙ(α ,β ) (z₁ )Qₙ(α ,β ) (z₂ ). Various confluent limits of these expressions give Nicholson-type integrals and product formulas for general Gegenbauer, Laguerre, Bessel, and Hermite functions. These results are summarized in the present paper. Derivations and applications will be given elsewhere.