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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 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 formulas with the mixed products of two Jacobi’s theta functions. As applications, we also obtain many interesting identities of Jacobi’s theta functions.

The aim of this paper is to investigate a linear precoding scheme design for a multiple-input multiple-output two-way relay system with imperfect channel state information. The scheme design is simplified as an optimal problem with precoding matrix variables, which is deduced with the maximum power constraint at the relay station based on the minimum mean square error criterion. With channel feedback delay at both ends of the channel and the channel estimation errors being taken into account, we propose a matrix decomposition scheme and a joint iterative scheme to minimize the average sum mean square error. The matrix decomposition method is used to derive the closed form of the relay matrix, and the joint iterative algorithm is used to optimize the precoding matrix and the processing matrix. According to numerical simulation results, the matrix decomposition scheme reduces the system bit error rate (BER) effectively and the joint iterative scheme achieves the best performance of BER against existing methods.

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.

The friction is a key factor of rolling bearing performance, especially for those precision transmission fields. All the top rolling bearing manufactures provide their calculating models on friction torque. However, there is great difference between the calculation result and measured value. Two calculation methods on friction torque of four-point-contact ball bearing, considered different factors involving the calculation, are compared and researched for discovering the key influence factor. A specific test is development to verify the theoretical analysis and a precision control method on friction torque with lower cost in production is proposed.

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

The Stirling number of the second kind S(n, k) is the number of partitions of {1,2,…,n} into k parts and is given by the following explicit formula: (1) S(n,k)=1k!∑j=0k(−1)k−j(kj)jn. Below we state...

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.

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 identities of theta functions are also shown.

In the paper, the authors prove that the functions |?(i)(ex)| for i ? N are subadditive on (ln?i,?) and superadditive on (-?, ln ?i), where ?i ? (0,1) is the unique root of equation 2|?(i)(?)|= ?(i)(?2)|.

Recently, Zhu gave an alternating circular summation of theta functions (see [Appl. Anal. Discrete Math. 6 (2012), 114–125]). In this paper, we further extend Zhu's results. Some applications are also shown.

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.

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

Carlitz firstly defined the $q$-Bernoulli and $q$-Euler polynomials [\textit{Duke Math. J.}, \textbf{15} (1948), 987--1000]. Recently, M. Cenkci and M. Can [\textit{Adv. Stud. Contemp. Math.}, \textbf{12} (2006), 213--223], J. Choi, P. J. Anderson and H. M. Srivastava [ \textit{Appl. Math. Comput.}, {\bf199} (2008), 723--737] further defined the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials. In this paper, we show the generating functions and basic properties of the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials, and obtain some relationships between the $q$-Apostol-Bernoulli and $q$-Apostol-Euler polynomials which are the corresponding $q$-extensions of some known results. Some formulas in series of $q$-Stirling numbers of the second kind are also considered.

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 inequalities for bounding the ratio of two gamma functions are discovered.

Completely monotonic and related functions are important function classes inmathematical analysis. It was Bernstein [1] who in 1914 first introduced the notion of completely monotonic function. This year we celebrate its 100th anniversary. In 1921, Hausdorff [2] gave the notion of completely monotonic sequence, which is related to the notion of completely monotonic function. Widder [3] in 1931 introduced the notion of minimal completely monotonic sequence. The terminology: logarithmically completely monotonic function, was first used in [4] in 1988. In 1989, the authors [5] introduced the notion of strongly completelymonotonic function. In 2009, the authors [6] introduced the notion of strongly logarithmically completely monotonic function. Also, the terminology: almost strongly completely monotonic function, was introduced in [6] in order to simplify the statement of the main results of the article [6]. In 2012, the terminology of almost completely monotonic function was defined and used in [7] in order to simplify the statement of the main results of the article [7]. Researchers did a lot of investigations, in both theory and applications, on these functions. More recently, for example, the authors [6] showed that a strongly logarithmically completely monotonic function must be almost strongly completely monotonic. Also, in [6], the following fact that a strongly logarithmically completely monotonic function cannot be strongly completely monotonic, or, equivalently, a strongly completely monotonic function cannot be strongly logarithmically completely monotonic was established. Figure 1 illustrates the inclusion relationships among these classes of functions on (0,∞). In these inclusion relationships, the fact that a logarithmically completely monotonic function must be completely monotonic was, in fact, first proved in [8] although the author did not use the terminology of logarithmically completelymonotonic function; all others are from [6] or directly from their definitions. Also note that in [6] counter examples were presented to show that some function classes are not included in someother function classes or to show that the intersection sets of two function classes are not empty. We invited investigators to contribute original research articles as well as comprehensive review articles to this special issue which will stimulate the continuing efforts to understand these functions.This special issuemainly focuses on the applications, in all fields, of completelymonotonic and related functions. The topics included in this special issue are as follows

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 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 of functions involving ratios of two gamma or $q$-gamma functions obtained by Bustoz, Ismail, Lorch, Muldoon, and other mathematicians.

In this paper, we provide some new integral inequalities of Qi type on time scales by using elementary analytic methods. Mathematics subject classification (2010): Primary 26E70; Secondary 26D15, 93C70.

In this paper, a spectral element method (SEM) is applied to simulate the axisymmetric flow in Czochralski crystal growth. The coordinate singularity for 1/r at r=0 is avoided by using the Gauss-Radau type quadrature points for the spatial discretization. The SEM solver is validated by its application to the benchmark problems suggested by Wheeler [1]. The stream function is solved by an artificial Poisson equation with the present method. The results show that it agrees well with available data in the literatures.

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 polynomials of higher order, that is, the so-called Apostol–Genocchi polynomials of higher order. For these generalized Apostol–Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol–Bernoulli polynomials and the Apostol–Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631–642] and pose two open problems on the subject of our investigation.

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.

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 Bernoulli and Euler polynomials and related known results.

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

In the article, the noted 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 function ln(x + y) are given. These results generalize A. Sard’s formula in numerical integration.

Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators (BEQPO's), we present the analytic expression of arbitrary matrix elements for BEQPO's. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO's.

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 Hurwitz zeta function. We also show some special cases, which include the corresponding results of Luo, Garg, Srivastava, Ozden, and Ozarslan etc.

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 main object of the present sequel to these earlier works is to derive several general properties and relationships involving the Apostol-Bernoulli and Apostol-Euler polynomials. Some of these general results can indeed be suitably specialized in order to deduce the corresponding properties and relationships involving the (generalized) Bernoulli and (generalized) Euler polynomials. Other relationships associated with the Stirling numbers of the second kind are also considered.

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 numbers and polynomials of higher order. We establish their elementary properties, derive several explicit representations for them in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) Zeta function, and deduce their special cases and applications which are shown here to lead to the corresponding results for the classical Bernoulli numbers and polynomials of higher order.

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 concepts of Bernoulli numbersBn, Bernoulli polynomialsBn(x), and the generalized Bernoulli numbersBn(a,b)are generalized to the oneBn(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 betweenBn,Bn(x),Bn(a,b), andBn(x;a,b,c)are established.