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In the paper, by Descartes' rule of signs and other techniques, the authors present the negativity, zeros, and extreme values of the single-variable polynomials \(\begin{align*} G(t)&=5 t^{43}-218 t^{30}+720 t^{17}-455 t^{13}-52,\\ H(t)&=5t^{29}\sum_{\ell=0}^{12}(13-\ell)t^\ell-t^{16}\sum_{\ell=0}^{12}(2704-213 \ell)t^{\ell}\\ &\quad-169t^{13}\sum_{\ell=0}^{2} (7+3\ell) t^\ell-52\sum_{\ell=0}^{12}(\ell+1)t^{\ell},\\ J(t)&=43 t^{30}-1308 t^{17}+2448 t^4-1183,\\ K(t)&=43t^{17}\sum_{k=0}^{12}t^k-1265t^4\sum_{k=0}^{12}t^{k} +1183\sum_{k=0}^{3}t^k. \end{align*}\)

By virtue of two approaches, the authors prove that the function \(G(t)=5 t^{43}-218 t^{30}+720 t^{17}-455 t^{13}-52\) is negative on \([0,1)\) and \(G(1)=0\).

International audience; In the paper, the authors introduce a notion "multivariate exponential polynomials" which generalize exponential numbers and polynomials, establish explicit formulas, inversion formulas, and recurrence relations for multivariate exponential polynomials in terms of the Stirling numbers of the first and second kinds with the help of the Fàa di Bruno formula, two identities for the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds, construct some determinantal inequalities and product inequalities for multivariate exponential polynomials with the aid of some properties of completely monotonic functions and other known results, derive the logarithmic convexity and logarithmic concavity for multivariate exponential polynomials, and finally find an application of multi-variate exponential polynomials to white noise distribution theory by confirming that multivariate exponential polynomials satisfy conditions for sequences required in white noise distribution theory.

In this paper we investigate the monotonicity properties related to the ratio of gamma functions, from which some related asymptotics and inequalities are established. Some special cases also confirm the conjectures of C.-P. Chen [Monotonicity properties, inequalities, and asymptotic expansions associated with the gamma function. Appl. Math. Comput. 283 (2016), 385--396.]., 10 pages, accepted for publication in Mediterranean Journal of Mathematics

In this paper we define the q-Laguerre type polynomials Un(x; y; z; q), which include q-Laguerre polynomials, generalized Stieltjes-Wigert polynomials, little q-Laguerre polynomials and q-Hermite polynomials as special cases. We also establish a generalized q-differential operator, with which we build the relations between analytic functions and Un(x; y; z; q) by using certain q-partial differential equations. Therefore, the corresponding conclusions about q-Laguerre polynomials, little q-Laguerre polynomials and q-Hermite polynomials are gained as corollaries. As applications, some generating functions and generalized Andrews-Askey integral formulas are given in the final section.

We study the condition on expanding an analytic several variables function in terms of products of the homogeneous generalized Al-Salam-Carlitz polynomials. As applications, we deduce bilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. We also gain multilinear generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials. Moreover, we obtain generalizations of Andrews-Askey integrals and Ramanujan q-beta integrals. At last, we derive U(n + 1) type generating functions for the homogeneous generalized Al-Salam-Carlitz polynomials.

In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

International audience; In the paper, the authors extend a function arising from the Bernoulli trials in probability and involving the gamma function to its largest ranges, find logarithmically complete monotonicity of these extended functions, and, in light of logarithmically complete monotonicity of these extended functions, derive some inequalities for multinomial coefficients and multivariate beta functions. These results recover, extend, and generalize some known conclusions.

In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in [J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials.

In this paper, we show how to deduce Askey–Wilson type integrals by the method of q-difference equation. In addition, we construct the relation between bilinear generating functions and solutions of q-difference equation. More over, we generalize Bailey's ψ 6 6 summation from the perspective of q-integral by the method of q-difference equation. At last, we deduce U ( n + 1 ) type generating functions for Al-Salam–Carlitz polynomials by the method of q-difference equation.

Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain U(n+1) type generating functions for Cigler’s polynomials.

International audience; In the paper, after concisely surveying some closed formulas and applications of special values of the Bell polynomials of the second kind for some special sequences and elementary functions, the authors newly establish some closed formulas for some special values of the Bell polynomials of the second kind.

In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.

In the paper, by virtue of (1) the Stirling inversion theorem and the binomial inversion theorem, (2) the Faa di Bruno formula and two identities for the Bell polynomials of the second kind, (3) a formula of higher order derivative for the ratio of two differentiable functions, the authors (1) present two explicit formulas, a determinantal expression, and a recursive relation for a sequence of unnamed polynomials, (2) derive two identities connecting the sequence of unnamed polynomials with the Bell polynomials, (3) recover a known identity connecting the sequence of unnamed polynomials with the Bell polynomials.

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

In the paper, the authors introduce a new concept of $r$-$\varphi$-preinvex functions and establish some new integral inequalities of Hermite-Hadamard type.

In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.

In the paper, the author present an inequality for bounding the harmonic number in terms of the hyperbolic cosine.

In the paper, some new integral inequalities are presented by using analytic methods.

Summary: In the paper, the famous Hermite-Hadamard integral inequality for convex functionsis generalized to and refined as ones for n-time differentiable functions which are s-convex in thesecond sense, and some known results are improved. 1 Introduction It is common knowledge in mathematical analysis that a function f: I R !R is saidto be convex on I6= ;iff( x+(1 )y) f(x)+(1 )f(y) (1.1)for all x;y2Iand 2[0;1]; If the inequality (1.1) reverses, then the function fis saidto be concave on I.The most important inequality in the theory of convex functions should be the famousHermite-Hadamard inequality (1.2) below. If fis a convex function on [a;b], then0 1b aZ ba f(t)dt f a+b2 f(a)+f(b)21b aZ ba f(t)dt: (1.2)If fis concave on [a;b], then the inequality (1.2) is reversed.In [11], Hermite-Hadamard inequality (1.2) was refined as follows. If f(x) is differ-entiable on [a;b] such that jf 0 (x)j q for q 1 is convex on [a;b], thenZb f(a)+f(b)21b a af(x)dx 1 b a4 jf 0 (a)j q +jf 0 (b)j

In this article, van der Corput's inequality is generalized by using the well known Euler-Maclaurin sum formula and other analytic techniques.

In this paper, two classes of functions, involving a parameter and the classical Euler gamma function, and two functions, involving the classical Euler gamma function, are verified to be logarithmically completely monotonic in or ; some inequalities involving the classical Euler gamma function are deduced and compared with those originating from certain problems of traffic flow, due to J. Wendel and A. Laforgia, and relating to the well known Stirling's formula.

In the article, a general double refinement of Jordan's inequality, for n∈ℕ, is established, where the coefficients α k and β k defined by recursing formulas (9) and (10) are the best possible. As an application, L. Yang's inequality is refined.

van der Corput's inequality is extended and refined by using Euler-Maclaurin formula and other analytic techniques.

An upper bound of a function with two independent variables is derived.

This is a survey and expository article. Some new developments on refinements, generalizations, and applications of Jordan's inequality and related problems, including some results about Wilker-Anglesio's inequality, some estimates for three kinds of complete elliptic integrals, and several inequalities for the remainder of power series expansion of the exponential function, are summarized.