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We obtain new results on increasing, decreasing, and hill-shape sequences ofreal numbers by rational functions and polynomials with positive integercoefficients in some generality. Unimodal sequences have these sequences ofreal numbers. Then, polynomials give unimodal sequences by suitablelog-concavities. Also, rational functions extend polynomials. In thismanuscript, we introduce a notion of merged-log-concavity of rational functionsand study the variation of unimodal sequences by polynomials with positiveinteger coefficients. Loosely speaking, the merged-log-concavity of rationalfunctions extends Stanley's $q$-log-concavity of polynomials by Young diagramsand Euler's identities of $q$-Pochhammer symbols. To develop a mathematicaltheory of the merged-log-concavity, we discuss positivities of rationalfunctions by order theory. Then, we give explicit merged-log-concave rationalfunctions by $q$-binomial coefficients, Hadamard products, and convolutions,extending the Cauchy-Binet formula. Also, Young diagrams yield unimodalsequences by rational functions and the merged-log-concavity. Then, we studythe variation of these unimodal sequences by critical points, which arealgebraic varieties in a suitable setting. In particular, we extend $t$-powerseries of $q$-Pochhammer symbols $(-t;q)_{\infty}$ and $(t;q)_{\infty}^{-1}$ inEuler's identities by the variation of unimodal sequences and polynomials withpositive integer coefficients. Also, we consider eta products and quantumdilogarithms. This gives the golden ratio of quantum dilogarithms as a criticalpoint. In statistical mechanics, we discuss grand canonical partition functionsof some ideal boson-fermion gases with or without Casimir energies (Ramanujansummation). This gives statistical-mechanical phase transitions by themathematical theory of the merged-log-concavity and critical points such as thegolden ratio.

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