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Unimodality preservation by ratios of functional series and integral transforms

Dmitrii Karp, Anna Vishnyakova, Yi Zhang

TL;DR

The paper develops a unified framework based on sign-regular kernels to guarantee unimodality preservation in the ratio $\dfrac{A(x)}{B(x)}$ of general functional series and in the ratio of integral transforms. It proves two main theorems: if the coefficient ratios $\{a_k/b_k\}$ are unimodal and the kernel is SR$_3$, then $F(x)=A(x)/B(x)$ is unimodal with monotonicity inherited or reversed according to the SR-signs; an analogous result holds for integral transforms with a sign-regular kernel and positive weight. The work collects a broad class of kernels (including Dirichlet, gamma, incomplete gamma, beta, $q$-Pochhammer, Laguerre, and Bessel-type kernels) and shows they yield unimodality results for factorial, inverse factorial, and various transform-based series, enabling wide applicability. Applications to ratios of generalized hypergeometric functions and Nuttall Q-functions illustrate practical impact, and the paper closes with conjectures on Bessel-function ratios and a discussion of how sign-regularity guides unimodality and monotonicity in these ratios.

Abstract

An elementary, but very useful lemma due to Biernacki and Krzyż (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity.

Unimodality preservation by ratios of functional series and integral transforms

TL;DR

The paper develops a unified framework based on sign-regular kernels to guarantee unimodality preservation in the ratio of general functional series and in the ratio of integral transforms. It proves two main theorems: if the coefficient ratios are unimodal and the kernel is SR, then is unimodal with monotonicity inherited or reversed according to the SR-signs; an analogous result holds for integral transforms with a sign-regular kernel and positive weight. The work collects a broad class of kernels (including Dirichlet, gamma, incomplete gamma, beta, -Pochhammer, Laguerre, and Bessel-type kernels) and shows they yield unimodality results for factorial, inverse factorial, and various transform-based series, enabling wide applicability. Applications to ratios of generalized hypergeometric functions and Nuttall Q-functions illustrate practical impact, and the paper closes with conjectures on Bessel-function ratios and a discussion of how sign-regularity guides unimodality and monotonicity in these ratios.

Abstract

An elementary, but very useful lemma due to Biernacki and Krzyż (1955) asserts that the ratio of two power series inherits monotonicity from that of the sequence of ratios of their respective coefficients. Over the last two decades it has been realized that, under some additional assumptions, similar claims hold for more general series ratios as well as for unimodality in place of monotonicity. This paper continues this line of research: we consider ratios of general functional series and integral transforms and furnish natural sufficiency conditions for preservation of unimodality by such ratios. Numerous series and integral transforms appearing in applications satisfy our sufficiency conditions, including Dirichlet, factorial and inverse factorial series, Laplace, Mellin and generalized Stieltjes transforms, among many others. Finally, we illustrate our general results by exhibiting certain statements on monotonicity patterns for ratios of some special functions. The key role in our considerations is played by the notion of sign regularity.
Paper Structure (7 sections, 12 theorems, 57 equations)

This paper contains 7 sections, 12 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples
  4. Applications
  5. Acknowledgments.
  6. Sign regularity of some $q$-Pochhammer kernels
  7. Derivative asymptotics for factorial and inverse factorial series

Key Result

Theorem 2.2

Suppose for a sequence of real continuous functions $(\varphi_n)_{n=0}^\infty$ defined on a convex subset $I$ of $\mathbb{R}$ containing more than one point, the kernel $K(n,x)=\varphi_n(x)\in SR_r$ on $\mathbb{N}_0\times I$ for some $r\ge2$. Then for every real sequence $(c_n)_{n=0}^\infty$ having converges uniformly on all compact subsets of $I$, we have Moreover, if $S^{-}(f(x))_{x\in I}=S^{-

Theorems & Definitions (19)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Lemma 4.1
  • Theorem 4.2
  • ...and 9 more