Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases

Abstract

We obtain some new results on the unimodal sequences of the real values of rational functions by polynomials with positive integer coefficients. Thus, we introduce the notion of merged-log-concavity of rational functions. Roughly speaking, the notion extends Stanley's $q$-log-concavity of polynomials. We construct explicit merged-log-concave rational functions by $q$-binomial coefficients, Hadamard products, and convolutions, extending the Cauchy-Binet formula. Then, we obtain the unimodal sequences of rational functions by Young diagrams. Moreover, we consider the variation of unimodal sequences by critical points that separate strictly increasing, strictly decreasing, and hill-shape sequences among almost strictly unimodal sequences. Also, the critical points are zeros of polynomials in a suitable setting. The study above extends the $t$-power series of $(\pm t;q)_{\infty}^{\mp 1}$ to some extent by polynomials with positive integer coefficients and the variation of unimodal sequences. We then obtain the golden ratio of quantum dilogarithms ($q$-exponentials) as a critical point. Additionally, we consider eta products, generalized Narayana numbers, and weighted $q$-multinomial coefficients, which we introduce. In statistical mechanics, we discuss the grand canonical partition functions of some ideal boson-fermion gases with or without Casimir energies (Ramanujan summation). The merged-log-concavity gives phase transitions on Helmholtz free energies by critical points of the metallic ratios including the golden ratio. In particular, the phase transitions implies non-zero particle vacua from zero particle vacua as the temperature rises.