Interlacing of the zeros of contiguous hypergeometric functions

Abstract

It is well-known that hypergeometric functions satisfy first order difference-differential equations (DDEs) with rational coefficients, relating the first derivative of hypergeometric functions with functions of contiguous parameters (with parameters differing by integer numbers). However, maybe it is not so well known that the continuity of the coefficients of these DDEs implies that the real zeros of such contiguous functions are interlaced. Using this property, we explore interlacing properties of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials as particular cases).

Bounds for extreme zeros of some classical orthogonal polynomials

“…The polynomials S n−m−1 are the dual polynomials introduced by de Boor and Saff in [5] or equivalently, the associated polynomials analysed by Vinet and Zhedanov in [24]. Related results are proved by Segura in [21].…”

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

“…, [ n−1 2 ]}. Next, using (5) with λ replaced by λ + 1, followed by (6), and (5) with n replaced by n + 1, a straightforward calculation shows that…”

“…Indeed, it also follows immediately from Segura's result (cf. [6,Theorem 1]) that the (single) zero of the (linear) de Boor-Saff polynomial that completes the interlacing of the zeros of p n−1 with those of p n+1 is given by one of the coefficients in the three term recurrence relation satisfied by the orthogonal sequence { p n } ∞ n=0 . Since sets of interlacing points are useful as nodes of interpolation in quadrature formulas, the question arises as to whether Stieltjes interlacing extends to the zeros of p n and q m when m < n − 1 and { p n } ∞ n=0 and {q n } ∞ n=0 are different orthogonal sequences.…”

Zeros of pseudo-ultraspherical polynomials

“…80] uses the notation P n are real and simple, lie in the open interval (−1, 1), and are located symmetrically about the origin. Properties of the zeros of this one-parameter family of classical orthogonal polynomials have been extensively investigated and include inequalities satisfied by the zeros [10], bounds for extreme zeros [2,17] monotonicity [7,11,18] and convexity properties [8,22], and interlacing between different families [13,16,25]. As λ decreases below − 1 2 , the zeros of C (λ) n depart from the interval (−1, 1) in pairs through the endpoints −1 and 1 with an additional pair of zeros leaving (−1, 1) as λ passes through −k + 1 2 , k = 1, 2, .…”