Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials
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Asymptotic Distribution of Zeros of a Certain Class of Hypergeometric Polynomials Addisalem Abathun1,2 · Rikard Bøgvad2
Received: 10 March 2015 / Revised: 21 May 2015 / Accepted: 8 June 2015 © Springer-Verlag Berlin Heidelberg 2015
Abstract We study the asymptotic behavior of the zeros of a family of a certain class −n, a , . . . , a
of hypergeometric polynomials A F B b , b 2, . . . , b A ; z , using the associated hyperge1 2 B ometric differential equation, as the parameters go to infinity. The curve configuration on which the zeros cluster is characterized as level curves associated with integrals on an algebraic curve. The algebraic curve is the hypergeometrc differential equation, using a similar approach to the method used in Borcea et al. (Publ Res Inst Math Sci 45(2):525–568, 2009). In a specific degenerate case, we make a conjecture that generalizes work in Boggs and Duren (Comput Methods Funct Theory 1(1):275–287, 2001), Driver and Duren (Algorithms 21(1–4):147–156, 1999), and Duren and Guillou (J Approx Theory 111(2):329–343, 2001), and present experimental evidence to substantiate it. Keywords Hypergeometric polynomials · Cauchy transform · Asymptotic zero measures · Hypergeometric differential equation Mathematics Subject Classification
33C05 · 33C20 · 31A35 · 34E05
Communicated by Stephan Ruscheweyh.
B
Addisalem Abathun [email protected] Rikard Bøgvad [email protected]
1
Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia
2
Department of Mathematics, Stockholm University, Stockholm, Sweden
123
A. Abathun, R. Bøgvad
1 Introduction The generalized hypergeometric function A F B is defined by the series AFB
(a) A ;z (b) B
= AFB
a1 , a2 , . . . , a A ;z b1 , b2 , . . . , b B
=
∞ k=0
j=A
j=1 (a j )k
zk k! j=1 (b j )k
j=B
(1.1)
where (α)k = α(α + 1) · · · (α + k − 1) = (α+k) (α) is the Pochhammer symbol. It has A numerator parameters a1 , a2 , . . . , a A , B denominator parameters b1 , b2 , . . . , b B , where A, B ∈ N and one variable z. Any of the quantities may be complex, but none of the denominator parameters is a non-positive integer. If one of the numerator parameters is a negative integer, say a1 = −n, n ∈ N, the series terminates and the series in (1.1) reduces to a polynomial of degree n in z, called a generalized hypergeometric polynomial. See [1,13] for references to hypergeometric functions. We are interested in the asymptotics of the zeros of the hypergeometric polynomial pn (z) = A F B (−n, a2 (n), . . . , a A (n); b1 (n), . . . , b B (n); z),
(1.2)
where ai (n), i = 1, . . . , A and b j (n), j = 1, . . . , B are complex-valued linear functions of n ∈ N. We show that, under some assumptions on the asymptotic behavior of the parameters, the zeros cluster on a finite union of curves, which may be identified as level curves of certain harmonic functions. We study the asymptotic limit of the zero-counting measure μn = (1/n) pn (z)=0 δz , where δz is the Dirac measure. The support of the limit μ = limn→∞ μn , which we know exi
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