Erratum: Product Formulas and Nicholson-Type Integrals for Jacobi Functions

Delsarte's approach to generalized translation operators via the solution of an associated Cauchy problem is used to derive the product formulas for the Bessel, Whittaker, and Jacobi functions in kernel form. As essential prerequisites, explicit representations of the corresponding Riemann functions are given for three cases. The main part of the paper deals with the Jacobi case, for which the derivation of the translation kernel is carried out explicitly. In the Bessel case, the results of Delsarte are covered, and in the Whittaker case, a generalization of previous results of the author on the Laguerre polynomial product formula is obtained.

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 $P_n^{(\alpha ,\beta )} (x)$ and $Q_n^{(\alpha ,\beta )} (x)$ 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, $Q_n^{(\alpha ,\beta )} (z_1 )Q_n^{(\alpha ,\beta )} (z_2 )$. 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.

We state certain product formulae for Jackson integrals associated with irreducible reduced root systems. The Jackson integral is defined here as a sum over any full-rank sublattice of the coweight lattice for the root system. In particular, a Weyl group symmetry classification of the Jackson integrals is done when they have an expression of a product of the Jacobi elliptic theta functions. Most of the product formulae investigated by Aomoto, Macdonald and Gustafson appear in the list of classifications. A new product formula for an F 4 root system is included in it.

We present a set of addition formulas for the Jacobi, Laguerre, Gegenbauer, and hyperbolic Bessel functions of the second kind, $Q_\nu ^{(\alpha ,\beta )} $, $N_\nu ^\alpha $, $D_\nu ^\alpha $ and $K_\nu $. These addition formulas are analogues of Koornwinder's addition formulas for $P_n^{(\alpha ,\beta )} $ and $L_n^\alpha $, and of Gegenbauer's addition formulas for $C_n^\alpha $ and $J_n $. The addition formulas are derived from a set of product formulas for the functions of the second kind derived previously by the author, and, conversely, can be integrated to give the product formulas.

Product formulas for Jacobian elliptic functions s n, s c, s d and their reciprocals are established. Applications to Legendre’s incomplete elliptic integral of the first kind and to the arc lemniscate sine function are given. Lower and upper bounds for elliptic functions and elliptic integrals mentioned above are also derived.

In this paper we present explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers $C_q\subset \mathbb R^q$ of type $B$. These formulas are related to continuous one-parameter families of probability-preserving convolution structures on $C_q\times\mathbb R$. These convolutions on $C_q\times\mathbb R$ are constructed via product formulas for the spherical functions of the symmetric spaces $U(p,q)/ (U(p)\times SU(q))$ and associated double coset convolutions on $C_q\times\mathbb T$ with the torus $\mathbb T$. We shall obtain positive product formulas for a restricted parameter set only, while the associated convolutions are always norm-decreasing. Our paper is related to recent positive product formulas of Rosler for three series of Heckman-Opdam hypergeometric functions of type BC as well as to classical product formulas for Jacobi functions of Koornwinder and Trimeche for rank $q=1$.

The main objective of this paper is to develop an infinite product formula for the ratio of consecutive prime numbers, using Jacobi elliptic functions.

We study Gelfand pairs for locally compact quantum groups. We give an operator algebraic interpretation and show that the quantum Plancherel transformation restricts to a spherical Plancherel transformation. As an example, we turn the quantum group analogue of the normaliser of SU(1,1) in $SL(2,\mathbb{C}$) together with its diagonal subgroup into a pair for which every irreducible corepresentation admits at most two vectors that are invariant with respect to the quantum subgroup. Using a $\mathbb{Z}_2$-grading, we obtain product formulae for little $q$-Jacobi functions.

Laguerre functions and representations of su(1,1)

In four cases it is already known that the product of two distinct Jacobian theta functions having the same variable z and the same nome q is a multiple of a single Jacobian theta function, with the multiple independent of z. The main purpose of the present note is to show that this property also applies in the remaining two cases.

In this paper we derive a q-analogue of the sampling theorem for Jacobi functions. We also establish a product formula for the nonterminating version of the q-Jacobi polynomials. The proof uses recent results in the theory of q-orthogonal polynomials and basic hypergeometric functions.