Turán type inequalities for regular Coulomb wave functions

Abstract

Turán, Mitrinović-Adamović and Wilker type inequalities are deduced for regular Coulomb wave functions. The proofs are based on a Mittag-Leffler expansion for the regular Coulomb wave function, which may be of independent interest. Moreover, some complete monotonicity results concerning the Coulomb zeta functions and some interlacing properties of the zeros of Coulomb wave functions are given.

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

“…This is clearly true in the case η = 0 and the resulting functions lead to the Bessel 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.…”

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

“…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…”

“…On the other hand, it is not difficult to use (20) and Corollary 3 in order to verify that the Hankel determinants for ℓ = 2, 3 read det H (2) n (ν) = 2 −2n(n+2) (n + 1)(n + ν + 1) n (ν) for n ∈ N, ℓ ∈ {0, 1}, and ν > −1, one can also use the formula (15) to prove that det H (ℓ) n (ν), as function of ν, is completely monotone for ν > −1, with n ∈ N and ℓ ∈ {0, 1} fixed. This can be easily checked by using the fact that a product of completely monotone functions is a completely monotone function.…”

“…This can be easily checked by using the fact that a product of completely monotone functions is a completely monotone function. Similarly, one can use (21) to verify the complete monotonicity of det H (2) n (ν) for ν > −1, with n ∈ N fixed. It seems that the same holds true even for ℓ ≥ 3, however, the verification would require a more sophisticated analysis.…”

Zeros of a cross-product of the Coulomb wave and Tricomi hypergeometric functions

“…) . Now, we shall use the Mittag-Leffler expansion of regular Coulomb wave function (obtained from its infinite product representation, see [1,7,8] for…”