[Paper Review] Tur\'an type inequalities for Tricomi confluent hypergeometric functions

Árpád Baricz, Mourad E. H. Ismail|arXiv (Cornell University)|Oct 21, 2011

Mathematical functions and polynomials14 references4 citations

TL;DR

This paper establishes sharp two-sided Turán type inequalities for Tricomi confluent hypergeometric functions and parabolic cylinder functions using integral representations derived from the infinite divisibility of the Fisher-Snedecor F distribution. The key contribution is a complete monotonicity result for Turán determinants of Tricomi functions, which improves upon earlier work by Ismail and Laforgia (2007).

ABSTRACT

Some sharp two-sided Tur\\'an type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions are deduced. The proofs are based on integral representations for quotients of parabolic cylinder functions and Tricomi confluent hypergeometric functions, which arise in the study of the infinite divisibility of the Fisher-Snedecor F distribution. Moroever, some complete monotonicity results are given concerning Tur\\'an determinants of Tricomi confluent hypergeometric functions. These complement and improve some of the results of Ismail and Laforgia [23].

Motivation & Objective

• To derive sharp two-sided Turán type inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions.
• To extend known results on Turán inequalities to special functions arising in probability and mathematical physics.
• To establish complete monotonicity properties of Turán determinants for Tricomi functions, improving upon prior work by Ismail and Laforgia.
• To connect these inequalities to the infinite divisibility of the Fisher-Snedecor F distribution through integral representations.
• To generalize results on determinants of modified Bessel and hypergeometric functions using multivariate integral formulas.

Proposed method

• Utilizes an integral representation formula from [22] for the ratio of parabolic cylinder functions to derive Turán inequalities.
• Applies a second integral representation from [22] specific to Tricomi confluent hypergeometric functions, linked to the infinite divisibility of the F distribution.
• Employs differential recurrence relations for parabolic cylinder functions to express the Turánian in terms of the function and its derivative.
• Derives a general determinant formula involving multivariate integrals over the cube $[-1,1]^{n+1}$ with weight functions involving $ (t_j^2 - 1)^{a - 1/2} $ and Vandermonde-like products.
• Uses the structure of the integral representation to prove complete monotonicity of Turán determinants by analyzing the positivity and monotonicity of the integrand.
• Applies known results on absolute and complete monotonicity of integrals with positive kernels to establish the monotonicity of the determinant expressions.

Experimental results

Research questions

• RQ1 What are the sharp bounds for the Turánian $ D_{-a}^2(x) - D_{-a-1}(x)D_{-a+1}(x) $ for parabolic cylinder functions?
• RQ2 How can integral representations of Tricomi confluent hypergeometric functions be used to derive Turán type inequalities?
• RQ3 What is the complete monotonicity status of the Turán determinant for Tricomi functions, and how does it improve upon Ismail and Laforgia (2007)?
• RQ4 Can multivariate integral formulas be used to prove monotonicity properties of higher-order Turán determinants?
• RQ5 What is the connection between the infinite divisibility of the Fisher-Snedecor F distribution and Turán inequalities for special functions?

Key findings

• A sharp Turán type inequality is proven for parabolic cylinder functions: $ 0 < D_{-a}^2(x) - D_{-a-1}(x)D_{-a+1}(x) ≤ \mu_a $, where $ \mu_a = \frac{\pi}{2^a}\left[\frac{1}{\Gamma^2\left(\frac{a+1}{2}\right)} - \frac{1}{\Gamma\left(\frac{a}{2}\right)\Gamma\left(\frac{a}{2}+1\right)}\right] $, with equality at $ x = 0 $.
• The Turánian for Tricomi confluent hypergeometric functions is shown to be completely monotonic in $ x $ for $ a > 0 $, extending results from Ismail and Laforgia (2007).
• A general formula is derived for the $ (n+1) \times (n+1) $ Turán determinant of functions with integral representations, expressed as a multivariate integral over $ [-1,1]^{n+1} $ with a positive kernel.
• The determinant of the non-central chi distribution's density functions is shown to be completely monotonic for $ a > 1/2 $, with an explicit integral representation.
• The paper provides a sharp Turán inequality for the non-central chi density: $ 0 < \chi_{2a+1,\tau}^2(x) - \chi_{2a,\tau}(x)\chi_{2a+2,\tau}(x) < \left[1 - \frac{\Gamma^2\left(a+\frac{1}{2}\right)}{\Gamma(a)\Gamma(a+1)}\right]\chi_{2a+1,\tau}^2(x) $, valid for all $ a, \tau, x > 0 $.
• The method establishes absolute and complete monotonicity of determinants whose entries are probability density functions, such as the non-central chi distribution, by analyzing the sign and monotonicity of the integrand in multivariate integrals.