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In this paper, we introduce a new unified family of polynomials including unified Apostol–Bernoulli, Euler and Genocchi polynomials, see Luo and Srivastava (2006) and a unified family of polynomials defined by Ozden et al. (2010). Also, by applying the Mellin transformation for the generating function of the new unified family of polynomials, we deduced a unified class of zeta function. Finally, we derive some relationships between our new unified family of polynomials and some types of generalized polynomials. [ABSTRACT FROM AUTHOR]

This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski–Grüss inequalities in . In the last note, we propose an improvement of the Ostrowski–Grüss inequality which involves knots where is an arbitrary numbers. More precisely, suppose that , and are arbitrary sequences with and . The main result of the present paper is to estimate in terms of either or . Unlike the standard Ostrowski–Grüss inequality and its known variants which basically estimate in terms of a correction term as a linear polynomial of x and some derivatives of f, our estimate allows us to freely replace and the correction term by using knots and . As far as we know, this is the first result involving the Ostrowski–Grüss inequality with three sequences of parameters. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to deal with applications of umbral calculus on fermionic p-adic integral on . From those applications, we derive some new identities on Genocchi numbers and polynomials. Moreover, a systemic study of the class of Sheffer sequences on the generating function of Genocchi polynomials are given. [Copyright &y& Elsevier]

Abstract: In this paper, we mainly consider the systems of difference equations where each of the sequences and represents either the sequence or the sequence , with nonzero real initial values and . Then we solve fourteen out of sixteen possible systems. It is noteworthy to depict that the solutions are presented in terms of Fibonacci numbers for twelve systems of these fourteen systems. [Copyright &y& Elsevier]

Abstract: In this paper, by considering the circulant matrix , whose entries are the generalized k-Horadam numbers, we present a new generalization to compute determinants and inverses of . In fact the results in here are the most general statements to obtain the inverses and determinants in such matrices having the elements of all second order sequences. [Copyright &y& Elsevier]

Abstract: This paper studies a sequence-dependent scheduling problem incorporating order delivery, motivated by satellite imaging scheduling. A set of jobs is to be processed on a single machine and each job belongs to a specific group. The completion time of a group is the moment when all jobs belonging to this group are completed. The problem is to determine a processing sequence of the jobs such that the sum of weighted completion times over all groups is minimized. We present a binary integer program to formulate the studied problem and then develop an O(n 22 n ) dynamic programming algorithm for determining optimal solutions. To produce approximate solutions within an acceptable time, we design a tabu search algorithm, an iterated local search algorithm and a genetic algorithm. Computational experiments are conducted to study the performance of the integer program and the solution algorithms. Numerical statistics suggest that the binary integer program can reach optimal solutions faster than the integer program existing in the literature, and the iterated local search algorithm outperforms other approaches when the number of jobs increases. [Copyright &y& Elsevier]

Abstract: This article deals with the introduction of Appell type sets of polynomials and associated numbers. The 2-iterated Appell polynomials are introduced and the Bernoulli and Euler based Appell polynomials are deduced as their particular cases. Different sets of polynomials, namely 2-iterated Bernoulli and Euler, Bernoulli–Euler (or Euler–Bernoulli) and their related numbers are considered. The operational relations between these families and Appell polynomials are used on the results of the Bernoulli and Euler polynomials to obtain the results for the corresponding mixed polynomials. [Copyright &y& Elsevier]

Abstract: It is shown that the next system of k difference equationswhere , and initial values , are real numbers, and where is defined by , , can be solved in closed form in an elegant way. Some applications of obtained formulas, for the case when the sequences and are k-periodic are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we compute the spectral norms of the matrices related with integer sequences and we give two examples related with Fibonacci, Lucas, Pell and Perrin numbers. [Copyright &y& Elsevier]

Abstract: It is shown that the system of difference equationswhere , the sequences , , and the initial values , are real numbers can be solved in closed form. Asymptotic behavior of well-defined solutions of the system for the case when all the sequences and are constant is studied in detail. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to study on the Genocchi polynomials of higher order on P, the algebra of polynomials in the single variable x over the field of characteristic zero and , the vector spaces of all linear functional on P. By using the action of a linear functional L on a polynomial Sheffer sequences and Appell sequences, we obtain some fundamental properties of the Genocchi polynomials. Furthermore, we give relations between, the first and second kind Stirling numbers, Euler polynomials of higher order and Genocchi polynomials of higher order. [Copyright &y& Elsevier]

Abstract: Let a 0, a 1,…, a r−1 be positive numbers and define a sequence {q m }, with initial conditions q 0 =0 and q 1 =1, and for all m ⩾2, q m = a t q m−1 + q m−2 where m ≡ t(mod r). For r =2, the author called the sequence {q m } as the generalized Fibonacci sequences and studied it in . But, it remains open to find a closed form of the generating function for general {q m }. In this paper, we solve this open problem, that is, we find a closed form of the generating function for {q m }in terms of the continuant. [Copyright &y& Elsevier]

Abstract: In this paper we study the asymptotic behavior and the periodicity of the positive solutions of the nonautonomous difference equation:where A n is a positive bounded sequence, p, q ∈(0,∞) and x −1, x 0 are positive numbers. [Copyright &y& Elsevier]

Abstract: In this paper, we study the sums of k-Fibonacci numbers with indexes in an arithmetic sequence, say for fixed integers a and r. This enables us to give in a straightforward way several formulas for the sums of such numbers. [Copyright &y& Elsevier]

Abstract: In this paper, using elementary symmetric function and complete homogeneous symmetric function, we obtain a determinant formula for the k-generalized Fibonacci sequence. The relationship between the k-generalized Fibonacci sequence and the order-k linear homogeneous recurrence relation has been investigated, and a general solution for the latter is also derived. Furthermore, we obtain an explicit expression for the elements in the nth power of the companion matrix in terms of k-generalized Fibonacci numbers. [Copyright &y& Elsevier]