Multiplication Theorem Research Articles

Some symmetric identities on higher order q-Euler polynomials and multivariate fermionic p-adic q-integral on [formula omitted

In this paper we derive some identities of symmetry related to higher order q-Euler polynomials by using the multivariate fermionic p-adic q-integral on Zp. Furthermore, some of these identities are also related to the q-analogue of the alternating power sums and the multiplication theorem.

Improved Three-Way Split Formulas for Binary Polynomial and Toeplitz Matrix Vector Products

In this paper, we consider three-way split formulas for binary polynomial multiplication and Toeplitz matrix vector product (TMVP). We first recall the best known three-way split formulas for polynomial multiplication: the formulas with six recursive multiplications given by Sunar in a 2006 IEEE Transactions on Computers paper and the formula with five recursive multiplications proposed by Bernstein at CRYPTO 2009. Second, we propose a new set of three-way split formulas for polynomial multiplication that are an optimization of Sunar's formulas. Then, we present formulas with five recursive multiplications based on field extension. In addition, we extend the latter formulas to TMVP. We evaluate the space and delay complexities when computations are performed in parallel and provide a comparison with best known methods.

Multiplication theorems for multi-variable and multi-index Bessel functions

In this article, we derive multiplication theorems for the 3-variable 2-parameter Bessel functionsJn(x;y;z ; 1; 2) and 2-index 5-variable 3-parameter Bessel functions Jm;n(x;y;z;w;h ; 1; 2; 3) using the generating function method. Further, we derive multiplication theorems for functions related to Jn(x;y;z ; 1; 2) and Jm;n(x;y;z;w;h ; 1; 2; 3). Furthermore, we establish a multiplication theorem for N-index Bessel functions Jm1;m2:::::mN (x ).

Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications

The first aim of this paper is to construct new generating functions for the generalized {\lambda}-Stirling type numbers of the second kind, generalized array type polynomials and generalized Eulerian type polynomials and numbers, attached to Dirichlet character. We derive various functional equations and differential equations using these generating functions. The second aim is provide a novel approach to deriving identities including multiplication formulas and recurrence relations for these numbers and polynomials using these functional equations and differential equations. Furthermore, by applying p-adic Volkenborn integral and Laplace transform, we derive some new identities for the generalized {\lambda}-Stirling type numbers of the second kind, the generalized array type polynomials and the generalized Eulerian type polynomials. We also give many applications related to the class of these polynomials and numbers.

A reduction formula for the hypergeometric function using gamma random variables

We show that the hypergeometric function p F q is the mean of 0 F q with its last argument multiplied by the product of independent gamma random variables. We use this to express for α>0 and β>0 in terms of 1 F 1, where I p is the modified Bessel function. A second derivation gives the moments of the non-central chi-square random variable. Some related results are derived, including an analog of Gauss's duplication formula for p F q .

A NOTE ON THE TWISTED LERCH TYPE EULER ZETA FUNCTIONS

In this note, the <TEX>$q$</TEX>-extension of the twisted Lerch Euler zeta functions considered by Jang [Bull. Korean Math. Soc. 47 (2010), no. 6, 1181-1188] is further investigated, and the generalized multiplication theorem for the <TEX>$q$</TEX>-extension of the twisted Lerch Euler zeta functions is given. As applications, some well-known results in the references are deduced as special cases.

Local Invariants on Quotient Singularities and a Genus Formula for Weighted Plane Curves

In this paper we extend the concept of Milnor fiber and Milnor number of a curve singularity allowing the ambient space to be a quotient surface singularity. A generalization of the local {\delta}-invariant is defined and described in terms of a Q-resolution of the curve singularity. In particular, when applied to the classical case (the ambient space is a smooth surface) one obtains a formula for the classical {\delta}-invariant in terms of a Q-resolution, which simplifies considerably effective computations. All these tools will finally allow for an explicit description of the genus formula of a curve defined on a weighted projective plane in terms of its degree and the local type of its singularities.

Duplication formulae involving Jacobi theta functions and Gosper’s 𝑞-trigonometric functions

Using the q q -trigonometric definitions of Gosper, we devise a new q q -exponential function. Based on this concept, we derive a number of identities involving the Jacobi theta functions. These considerations lead to the answers to Gosper’s “mysteries”.

Normalized polynomials and their multiplication formulas

The aim of this paper is to prove multiplication formulas of the normalized polynomials by using the umbral algebra and umbral calculus methods. Our polynomials are related to the Hermite-type polynomials.

Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums

In recent years, symmetry properties of the Bernoulli polynomials and the Euler polynomials have been studied by a large group of mathematicians (He and Wang in Discrete Dyn. Nat. Soc. 2012:927953, 2012, Kim et al. in J. Differ. Equ. Appl. 14:1267-1277, 2008; Abstr. Appl. Anal. 2008, doi:11.1155/2008/914347, Yang et al. in Discrete Math. 308:550-554, 2008; J. Math. Res. Expo. 30:457-464, 2010). Luo (Integral Transforms Spec. Funct. 20:377-391, 2009), introduced the lambda-multiple power sum and proved the multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order. Ozarslan (Comput. Math. Appl. 2011:2452-2462, 2011), Lu and Srivastava (Comput. Math. Appl. 2011, doi:10.1016/j.2011.09.010.2011) gave some symmetry identities relations for the Apostol-Bernoulli and Apostol-Euler polynomials.In this work, we prove some symmetry identities for the Apostol-Bernoulli and Apostol-Euler polynomials related to multiple alternating sums.AMS Subject Classification: 11F20, 11B68, 11M35, 11M41.

On p-adic interpolating function associated with modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight α

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. Furthermore we derive witt's type formula and Distribution formula (Multiplication theorem) for modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. In section three, by applying Mellin transformation we define q-analogue of modified twisted q-l-functions of Dirichlet's type and also we deduce that it can be written as modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. Moreover we will find a link between modified twisted Hurwitz-zeta function and q-analogue of modified twisted q-l-functions of Dirichlet's type which yields a deeper insight into the effectiveness of this type of generalizations. In addition we consider q-analogue of partial zeta function and we derive behavior of the modified q-Euler L-function at s = 0. In final section, we construct p-adic twisted Euler q-L function with weight alpha and interpolate Dirichlet's type of twisted q-Euler polynomials with weight alpha at negative integers. Our new generating function possess a number of interesting properties which we state in this paper.

The Real Genus of &lt;i&gt;p&lt;/i&gt;-Groups

Let G be a finite group. The real genus p(G) is the minimum algebraic genus of any compact bordered Klein surface on which G acts. We show that a non-cyclic p-group G has real genus not congruent to (1 — p3) (mod 2p3) if and only if G is in one of 14 families of groups. The genus formula for each of these 14 families of groups is determined. A consequence of this classification is that almost all positive integers that are the real genus of a p-group are congruent to (1 p3) (mod 2p3). Finally, the integers that occur as the real genus of a p-group with Frattini-class 2 have density zero in the positive integers.

Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials

The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials.

Quantum groups and quantum cohomology

In this paper, we study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver Q. Using a geometric R-matrix formalism, we construct a Hopf algebra Y_Q, the Yangian of Q, acting on the cohomology of these varieties, and show several results about their basic structure theory. We prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for Y_Q/ equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of Y_Q. A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection. In the second part, we give an extended example of the general theory for moduli spaces of sheaves on C^2, framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization/ the corresponding R-matrix is closely related to the reflection operator in Liouville field theory. We show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, we obtain an action of the W-algebra W(gl(r)) on the equivariant cohomology of rank $r$ moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

The Remodeling Conjecture and the Faber–Pandharipande Formula

In this note, we prove that the free energies F g constructed from the Eynard–Orantin topological recursion applied to the curve mirror to \({\mathbb{C}^3}\) reproduce the Faber–Pandharipande formula for genus g Gromov–Witten invariants of \({\mathbb{C}^3}\) . This completes the proof of the remodeling conjecture for \({\mathbb{C}^3}\) .

New Bit Parallel Multiplier With Low Space Complexity for All Irreducible Trinomials Over $GF(2^{n})$

Koc and Sunar proposed an architecture of the Mastrovito multiplier for the irreducible trinomial f(x)=xn+xk+1, where k ≠ n/2 to reduce the time complexity. Also, many multipliers based on the Karatsuba-Ofman algorithm (KOA) was proposed that sacrificed time efficiency for low space complexity. In this paper, a new multiplication formula which is a variant of KOA presented. We also provide a straightforward architecture of a non-pipelined bit-parallel multiplier using the new formula. The proposed multiplier has lower space complexity than and comparable time complexity to previous Mastrovito multipliers' for all irreducible trinomials.

Design of 4x4 bit Vedic Multiplier using EDA Tool

The need of high speed multiplier is increasing as the need of high speed processors are increasing. A Multiplier is one of the key hardware blocks in most fast processing system which is not only a high delay block but also a major source of power dissipation. A conventional processor requires substantially more hardware resources and processing time in the multiplication operation, rather than addition and subtraction. This paper presents a high speed 4x4 bit Vedic Multiplier (VM) based on Vertically & Crosswise method of Vedic mathematics, a general multiplication formulae equally applicable to all cases of multiplication. It is based on generating all partial products and their sum in one step. The coding is done in VHDL (Very High Speed Integrated Circuit Hardware Descriptive Language) while the synthesis and simulation is done using EDA (Electronic Design Automation) tool XilinxISE12.1i. The combinational path delay of 4x4 bit Vedic multiplier obtained after synthesis is compared with normal multipliers and found that the proposed Vedic multiplier circuit seems to have better performance in

Functional definitions for q-analogues of Eulerian functions and applications

We explore a number of functional properties of the $q$-gamma function and a class of its quotients; including the $q$-beta function. We obtain formulas for all higher logarithmic derivatives of these quotients and give precise conditions on their sign. We prove how these and other functional properties, such as the multiplication formula or the asymptotic expansion, together with the fundamental functional equation of the $q$-gamma function uniquely define those functions. We also study reciprocal "relatives" of the fundamental $q$-gamma functional equation, and prove uniqueness of solution results for them. In addition, we also use a reflection formula of Askey to derive expressions relating the classical sine function and the number $\pi$ to the $q$-gamma function. Throughout we highlight the similarities and differences between the cases $0<q<1$ and $q>1$.

Meromorphic representations of products of complex arithmetic progressions

Given a complex arithmetic sequence, a + nd, where \({a,d \in \mathbb{C}}\), d ≠ 0, and \({n \in \mathbb{Z}^+}\), define \({\Pi^a_d(a+nd):= a(a+d)\cdots (a+nd)}\). At first \({\Pi_d^a}\) is only defined on the terms of the arithmetic sequence. In this article, \({\Pi_d^a}\) is extended to a meromorphic function on \({\mathbb{C} \setminus\{-d,-2d,\dots\}}\) which satisfies the functional equation, \({\Pi_d^a(z+d)=(z+d)\Pi_d^a(z)}\). This extension is represented in three ways: in terms of the classical \({\Pi}\) function; as a limit involving a Pochhammer-type symbol; as an infinite product involving a generalized Euler constant. The infinite product representation leads to a natural Multiplication Formula for the functions \({\Pi_d^a}\), which, in turn, provides an easy way to prove Gauss’s Multiplication Formula for the Γ function.

Multiplication formulas for Apostol-type polynomials and multiple alternating sums

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.