Orthogonal polynomials associated with Coulomb wave functions

Abstract

A class of orthogonal polynomials associated with Coulomb wave functions is introduced. These polynomials play a role analogous to that the Lommel polynomials do in the theory of Bessel functions. The measure of orthogonality for this new class is described explicitly. In addition, the orthogonality measure problem is also discussed on a more general level. Apart of this, various identities derived for the new orthogonal polynomials may be viewed as generalizations of some formulas known from the theory of Bessel functions. A key role in these derivations is played by a Jacobi (tridiagonal) matrix J L whose eigenvalues coincide with reciprocal values of the zeros of the regular Coulomb wave function F L (η, ρ). The spectral zeta function corresponding to the regular Coulomb wave function or, more precisely, to the respective tridiagonal matrix is studied as well.

Complementary Romanovski-Routh polynomials: From orthogonal polynomials on the unit circle to Coulomb wave functions

“…With such a knowledge, the author of [6] looks at some Turán type inequalities associated with these extended regular Coulomb wave functions F λ (η, w) and obtains also some information regarding the zeros of these functions. With a different objective, the authors of [30] study the orthogonal polynomials that follow from the extended three term recurrence relation, which they call orthogonal polynomials associated with Coulomb wave functions. For some other contributions regarding the functions F λ (η, w) for non-integer values of λ, and even complex values of the parameter λ, we cite, for example, [11,21] and references therein.…”

The Hurwitz-type theorem for the regular Coulomb wave function via Hankel determinants

“…, is absolutely convergent for k ≥ 2, L / ∈ −(N + 1)/2, and η ∈ C. Here we use the notation from [27] where ζ L is referred to as the spectral zeta function since the zeros of φ L (η, ·) are eigenvalues of a certain Jacobi operator. Although ζ L as well as ρ L,n depend also on η, the dependence is not explicitly designated for brevity.…”

“…To obtain the formula from the statement, we make use of the particular family of orthogonal polynomials studied in [27], for which the moment Hankel matrix coincides with H n (L, η) up to an unimportant multiplicative factor. These polynomials satisfies (11) with a n = (n + L + 1) 2 + η 2 (n + L + 1) 2…”

“…Remark 2. The functions ζ L (n), for n ≥ 2, appear as coefficients in the power series expansion of the logarithmic derivative of φ L (η, ρ) with respect to ρ, see, for example, the first unlabeled equation above [27,Eq. (77)].…”

Nevanlinna extremal measures for polynomials related to q−1-Fibonacci polynomials

“…, x m ) (22) which holds true for m, n ∈ Z + , n < m, see [34, Lemma 1]. By using (21), one easily verifies that the sequence {p n } ∞ n=0 determined by the second-order recurrence p n+1 = α n p n + β n p n−1 , for n ∈ Z + , and initial conditions p −1 = 0 and p 0 = 1, can be expressed with the aid of the function F, see, for example, the general formula [35,Eq. (19)].…”