Qiu-Ming Luo

Chongqing Normal University | CNU · Department of Mathematics

In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function $\frac{x^{M}}{(x+1)^{\lambda}_{n}}$ x M ( x + 1 ) n λ ( M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combina...

In this note, we give a short proof of a modular equation of Ramanujan–Selberg continued fraction given by Chan (Proc Am Math Soc 137:2849–2856, 2009).

In this paper, we give some extensions for Ramanujan’s circular summation formulas with the mixed products of two Jacobi’s theta functions. As applications, we also obtain many interesting identities of Jacobi’s theta functions.

In this paper, by constructing contour integral and using Cauchy’s residue theorem, we provide a novel proof of Chu’s two partial fraction decompositions.

In this paper, we give some extensions for Mortenson’s identities in series with the Bell polynomial using the partial fraction decomposition. As applications, we obtain some combinatorial identities involving the harmonic numbers.

In this paper, we give two Ramanujan-type circular summation formulas by applying the way of elliptic functions and the properties of theta functions. As applications, we obtain the corresponding imaginary transformation formulas for Ramanujan-type circular summations and some theta function identities.

In this paper, we give some extensions for Ramanujan's circular summation formula with the mixed products of two Jacobi's theta functions. As some applications, we also obtain many interesting identities of Jacobi's theta functions.

In 1915, Ramanujan stated the following formula ∫ 0 ∞ t x - 1 ( - a t ; q ) ∞ ( - t ; q ) ∞ d t = π sin π x ( q 1 - x , a ; q ) ∞ ( q , a q - x ; q ) ∞ , where 0 < q < 1 , x > 0 , and 0 < a < q x . The above formula is called Ramanujan’s beta integral. In this paper, by using q-exponential operator, we further extend Ramanujan’s beta integral. As s...

In this paper, we obtain a new identity using the partial fraction decomposition. As applications, some interesting binomial identities are also derived.

We define an elliptic extension of the Genocchi polynomials and obtain the sums of products for the elliptic Genocchi polynomials. The formulas of sums of products for the Genocchi polynomials are also derived.

In this paper we derive a recurrence formula for the q-beta integral using the q-Chu-Vandermonde formula and show some special cases and applications.

In this paper, we generalize Rice’s integral and give some applications. Some old and new binomial identities are obtained. MSC: 05A10, 05A19, 11B65.

Recently, a family of the Apostol-type polynomials was introduced by Luo and Srivastava (Appl. Math. Comput. 217:5702-5728 (2011)). In this paper, we further investigate the Apostol-type polynomials and obtain their unified multiplication formula and explicit representations in terms of the Gaussian hypergeometric function and the generalized Hurwi...

We generalized Ramanujan's circular summation formula and give an elementary proof of them. We also show some applications and obtain some new identities of theta functions.

In this paper, we investigate the elliptic analogues of the Apostol–Bernoulli and Apostol–Euler polynomials and obtain the closed expressions of sums of products for these elliptic type polynomials. Some interesting special cases are also shown.

In this paper, we obtain the circular summation formulas of the theta function [InlineEquation not available: see fulltext.] and show the corresponding alternating summations and inverse relations. Some applications are also considered. MSC:11F27, 33E05.

In the paper, the authors prove that the functions $|\psi^{(i)}(e^x)|$ for $i\in\mathbb{N}$ are subadditive on $(\ln\theta_i,\infty)$ and superadditive on $(-\infty,\ln\theta_i)$, where $\theta_i\in(0,1)$ is the unique root of equation $2|\psi^{(i)}(\theta)|=|\psi^{(i)}(\theta^2)|$.

In the paper, the authors review some inequalities and the (logarithmically) complete monotonicity concerning the gamma and polygamma functions and, more importantly, present a sharp double inequality for bounding the polygamma function by rational functions.

In this article we establish a sharp two-sided inequality for bounding the Wallis ratio. Some best constants for the estimation of the Wallis ratio are obtained. An asymptotic formula for the Wallis ratio is also presented. MSC: 11B65, 41A44, 05A10, 26D20, 33B15, 41A60.

In the article we present necessary and sufficient conditions for a function involving the logarithm of the gamma function to be completely monotonic and apply these results to bound the gamma function $\Gamma(x)$, the $n$-th harmonic number $\sum_{k=1}^n\frac1k$, and the factorial $n!$.

We show the basic properties and generating functions of the q-Apostol-Bernoulli polynomials. Some recursive formulas are derived in series of the q-power sums. We also provide a explicit relation between the q-Apostol-Bernoulli polynomials and q-Hurwitz-Lerch zeta function.

In the article, some Huygens and Wilker type inequalities involving trigonometric and hyperbolic functions are refined and sharpened.

We obtain some new generating functions for q -Hahn polynomials and give their proofs based on the homogeneous q -difference operator.

We obtain an expectation formula and give the probabilistic proofs of some summation and transformation formulas of q-series based on our expectation formula. Although these formulas in themselves are not the probability results, the proofs given are based on probabilistic concepts.

We reproduce the Fourier expansions for Bernoulli and Euler polynomials using the Lipschitz summation formula and obtain the unified integral representations for Bernoulli and Euler polynomials.

Basic properties are established and generating functions are obtained for the q -Apostol–Euler polynomials. We define q -alternating sums and obtain q -extensions of some formulas from Integral Transform. Spectr. Funct., 20, 377–391 (2009). We also deduce an explicit relationship between the q -Apostol–Euler polynomials and the q -Hurwitz–Lerch ze...

In the paper, after reviewing the history, background, origin, and applications of the functions $\frac{b^{t}-a^{t}}{t}$ and $\frac{e^{-\alpha t}-e^{-\beta t}}{1-e^{-t}}$, we establish sufficient and necessary conditions such that the special function $\frac{e^{\alpha t}-e^{\beta t}}{e^{\lambda t}-e^{\mu t}}$ are monotonic, logarithmic convex, loga...

Recently, Chan and Liu give a new formula for circular summation of theta functions [see On a new circular summation of theta functions. J Number Theory. 2010;130:1190–1196]. In this note, we further extend their formula and derive the corresponding imaginary transformation and alternating summation formulas. As some applications, some new identiti...

In this paper, we obtain some new circular summation formulas of theta functions using the theory of elliptic functions. As applications we also show some interesting identities of theta functions.

Carlitz firstly defined the q-Bernoulli and q-Euler polynomials [Duke Math. J., 15(1948), 987-1000]. Recently, M.Cenkci and M.Can [Adv. Stud. Contemp. Math., 12(2006), 213-223], J.Choi, P.J.Anderson and H. M. Srivastava [ Appl. Math. Comput., 199 (2008), 723-737] further defined the q-Apostol-Bernoulli and q-Apostol-Euler polynomials. In this paper...

As a generalization of 2D Bernoulli polynomials, neo-Bernoulli polynomials are introduced from a point of view involving the use of nonexponential generating functions. Their relevant recurrence relations, the differential equations satisfied by them and some other properties are obtained. Especially, we obtain the relationships between them and ne...

In the expository review and survey paper dealing with bounds for the ratio of two gamma functions, along one of the main lines of bounding the ratio of two gamma functions, the authors look back and analyze some known results, including Wendel's asymptotic relation, Gurland's, Kazarinoff's, Gautschi's, Watson's, Chu's, Kershaw's, and Elezovi\'c-Gi...

In the paper, the authors present monotonicity results of a function involving the inverse hyperbolic sine. From these, the authors derive some inequalities for bounding the inverse hyperbolic sine. MSC: 26A48, 26D05, 33B10.

In the paper, necessary and sufficient conditions are provided for a function involving the divided difference of two psi functions to be completely monotonic. Consequently, a class of inequalities for sums are presented, the logarithmically complete monotonicity of a function involving the ratio of two gamma functions are derived, and two double i...

A family of the Apostol-type polynomials was introduced and investigated recently by Luo and Srivastava (see (Appl. Math. Comput. 217:5702-5728, 2011)). In this paper, we study this polynomial family on P , the algebra of polynomials in a single variable x over all linear functional on P . By using the way of the umbral algebra, we obtain some fund...

In this paper, we study some properties of the generalized Apostol-type polynomials (see (Luo and Srivastava in Appl. Math. Comput. 217:5702-5728, 2011)), including the recurrence relations, the differential equations and some other connected problems, which extend some known results. We also deduce some properties of the generalized Apostol-Euler...

In this paper, we obtain some circular summation formulas of theta functions using the theory of elliptic functions and show some interesting identities of theta functions and applications. MSC: 11F27, 33E05, 11F20.

Recently, Tremblay, Gaboury and Fugère introduced a class of the generalized Bernoulli polynomials (see Tremblay in Appl. Math. Let. 24:1888-1893, 2011). In this paper, we introduce and investigate an extension of the generalized Apostol-Euler polynomials. We state some properties for these polynomials and obtain some relationships between the poly...

The authors provide a simple proof of Oppenheim’s double inequality relating to the cosine and sine functions. In passing, the authors survey this topic.

In the paper, the monotonicity and logarithmic convexity of Gini means and related functions are investigated.

We obtain the multiplication formulas for the ApostolGenocchi polynomials of higher order by using the generalized multinomial identity. We introduce the λ-multiple alternating sums and show some explicit recursive formulas of the Apostol-Genocchi polynomials of higher order. Apostol-Bernoulli numbers and polynomials of higher order, Apostol-Genocc...

Some new integral inequalities of Qi-type on time scales are provided by using elementary analytic methods.

In the paper, the authors analyze and compare two double inequalities for bounding the tangent function, reorganize the proof in C.-P. Chen and F. Qi (A double inequality for remainder of power series of tangent function, Tamkang J. Math. 34 (4), 351-355, 2003) by using the usual definition of Bernoulli numbers, and correct some errors on page 6, (...

In the survey paper, along one of several main lines of bounding the ratio of two gamma functions, the authors retrospect and analyse Wendel's double inequality, Kazarinoff's refinement of Wallis' formula, Watson's monotonicity, Gautschi's double inequality, Kershaw's first double inequality, and the (logarithmically) complete monotonicity results...

We investigate multiplication formulas for Apostol-type polynomials and introduce λ-multiple alternating sums, which are evaluated by Apostol-type polynomials. We derive some explicit recursive formulas and deduce some interesting special cases that involve the classical Raabe formulas and some earlier results of Carlitz and Howard.

In this paper, we generalize the Genocchi polynomials and investigate its q-analogue. Some interesting results and relationships are obtained.

We extend the Genocchi polynomials and investigate their Fourier expansions and integral representations. We obtain their formulas at rational arguments in terms of Hurwitz zeta function and show an explicit relationship with Gaussian hypergeometric functions. Some known results for the classical Genocchi polynomials are also deduced.

Recently, the authors introduced some generalizations of the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290–302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917–925]). The main object of this paper is to investigate an analogous generalization of the Genocchi po...

The main object of this paper is to give q-extensions of several explicit relationships of H. M. Srivastava and A. Pintér [Appl. Math. Lett. 17, No. 4, 375–380 (2004; Zbl 1070.33012)] between the Bernoulii and Euler polynomials. We also derive several other formulas in series of Carlitz’s q-Stirling numbers of the second kind.

We show some results for the q-Bernoulli and q-Euler polynomials. The formulas in series of the Carlitz's q-Stirling numbers of the second kind are also considered. The q-analogues of well-known formulas are derived from these results.

In this paper, we introduce the so-called lambda-Stirling numbers of the second kind and research its some elementary properties. We give an explicit relationship between the generalized Apostol-Bernoulli and Apostol-Euler polynomials interms of the lambda-Stirling numbers of the second kind.

We investigate Fourier expansions for the Apostol-Bernoulli and Apostol-Euler polynomials using the Lipschitz summation formula and obtain their integral representations. We give some explicit formulas at rational arguments for these polynomials in terms of the Hurwitz zeta function. We also derive the integral representations for the classical Ber...

We give some explicit relationships between the Apostol-Eulerpolynomials and generalized Hurwitz-Lerch Zeta function and obtainsome series representations of the Apostol-Euler polynomials ofhigher order in terms of the generalized Hurwitz-Lerch Zetafunction. Several interesting special cases are also shown.

This article obtains the multiplication formulas for the Apostol–Bernoulli and Apostol–Euler polynomials of higher order and deduces some explicit recursive formulas. The λ-multiple power sum and λ-multiple alternating sum that are clearly evaluated associated with the Apostol–Bernoulli and Apostol–Euler polynomials of higher order, respectively, a...

In this paper, by using the Lipschitz summation formula, we obtain Fourier expan-sions and integral representations for the Genocchi polynomials. Some other new and interesting results are also shown.

In this paper, we define the Apostol-Genocchi polynomials and q-Apostol-Genocchi polynomials. We give the generating function and some basic properties of q-Apostol-Genocchi polynomials. Several interesting relationships are also obtained.

In the article, the well-known Darboux's formula of functions with single variable is generalized to that of functions of two independent variables with integral remainder, some important special cases of Darboux's formula of functions with two variables are obtained, and some estimates of the integral remainders and Darboux's expansion of the func...

The purpose of this paper is to give analogous definitions of Apostol type (see T. M. Apostol (Pacific J. Math. 1 (1951), 161-167)) for the so-called Apostol-Euler numbers and polynomials of higher order. We establish their elementary properties, obtain several explicit formulas involving the Gaussian hypergeometric function and the Stirling number...

The object of the present note is to prove certain new explicit formulas for the Euler numbers and polynomials; these formulas involve the Stirling numbers of the second kind and the Gaussian hypergeometric function, respectively.

Recently, Srivastava and Pintér [1] investigated several interesting properties and relationships involving the classical as well as the generalized (or higher-order) Bernoulli and Euler polynomials. They also showed (among other things) that the main relationship (proven earlier by Cheon [2]) can easily be put in a much more general setting. The m...

The object of the present note is to prove a new explicit formulae for the Euler numbers of higher order.

The main object of this paper is to give analogous definitions of Apostol type (see [T.M. Apostol, On the Lerch Zeta function, Pacific J. Math. 1 (1951) 161–167] and [H.M. Srivastava, Some formulas for the Bernoulli and Euler polynomials at rational arguments, Math. Proc. Cambridge Philos. Soc. 129 (2000) 77–84]) for the so-called Apostol–Bernoulli...

This note presents a new proof of Gould's famous formula for the Bernoulli numbers.

In the present paper, we obtain two new formulas of the Apostol-Bernoulli polynomials (see On the Lerch Zeta function. Pacific J. Math., 1 (1951), 161–167.), using the Gaussian hypergeometric functions and Hurwitz Zeta functions respectively, and give certain special cases and applications.

The object of the present note is to prove certain new explicit formulae for the Euler num- bers and polynomials of higher order, these representations include the Stirling numbers of the second kind.

In this article, using the L'Hospital rule, mathematical induction, the trigonometric power formulas and integration by parts, some integral formulas for the improper integrals $\int_0^\infty\frac{\sin^{2m}(\alpha x)}{x^{2n}}\td x$ and $\int_0^\infty\frac{\sin^{2m+1}(\alpha x)}{x^{2n+1}}\td x$ are established, where $m\ge n$ are all positive intege...

In the article, K. Petr's formula of single integral is generalized to that of double integral, some important special cases and estimates of its remainder are established.

Using the L’Hospital rule, mathematical induction, the trigonometric power formulae and integration by parts, some integral formulae for the improper integrals ∫ 0 ∞ [sin 2m (αz)]/(x 2n )dx and ∫ 0 ∞ [sin 2m+1 (αz)]/(x 2n+1 )dx are established, where m≥n are all positive integers and α≠0.

In this article, using the de l’Hospital rule, mathematical induction, the trigonometric power formulae and integration by parts, some integral formulae for the improper integrals from the title are established, where m≥n are all positive integers and α≠0.

See http://dx.doi.org/10.1017/S0025557200173863

In this article, the convergence of the sequence graphics is proved, and some inequalities involving this sequence are established for a > 0. As byproduct, two identities involving irrational numbers are obtained. Two open problems are proposed.

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In this article, the convergence of the sequence a+a+⋯+a 3 3 3 ︸ n is proved, and some inequalities involving this sequence are established for a>0. As by-product, two identities involving irrational numbers are obtained. Two open problems are proposed.

The concepts of Bernoulli numbers Bn, Bernoulli polynomials Bn(x), and the generalized Bernoulli numbers Bn(a,b) are generalized to the one Bn(x;a,b,c) which is called the generalized Bernoulli polynomials depending on three positive real parameters. Numerous properties of these polynomials and some relationships between Bn, Bn(x), Bn(a,b), and Bn(...

The concepts of Euler numbers and Euler polynomials are generalized and some basic properties are investigated.

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An inequality of H. Minc and L. Sathre [Proc. Edinb. Math. Soc., II. Ser. 14, 41-46 (1964; Zbl 0124.01003)] is generalized as follows: Let n and m be natural numbers, k a nonnegative integer, then we have n+k n+m+k<(n+k)!/k! n (n+m+k)!k! n+m<1· From this, some corollaries are deduced. At last, an open problem is proposed.

An inequality of H. Minc and L. Sathre (Proc. Edinburgh Math. Soc. ${\bf 14}$(1964/65), 41-46) is generalized as follows: Let $n$ and $m$ be natural numbers, $k$ a nonnegative integer, then we have $${n+k\over n+m+k} From this, some corollaries are deduced. At last, an open problem is proposed.