Generating functions of the (<Emphasis Type="Italic">h,q</Emphasis>) extension of twisted Euler polynomials and numbers

By using p-adic q-deformed fermionic integral on ℤ _p , we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.      

Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions

In this paper, we systematically recover the identities for the q-eta numbers η_k and the q-eta polynomials η_k(x), presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler’s approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175–192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz’s q-Bernoulli polynomials -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.     

Multiple <Emphasis Type="Italic">q</Emphasis>-zeta functions and multiple <Emphasis Type="Italic">q</Emphasis>-polylogarithms

For every positive integer d we define the q-analog of multiple zeta function of depth d and study its properties, generalizing the work of Kaneko et al. who dealt with the case d=1. We first analytically continue it to a meromorphic function on ℂ ^d with explicit poles. In our Main Theorem we show that its limit when q ↑1 is the ordinary multiple zeta function. Then we consider some special values of these functions when d=2. At the end of the paper we also propose the q-analogs of multiple polylogarithms by using Jackson’s q-iterated integrals and then study some of their properties. Our definition is motivated by those of Koornwinder and Schlesinger although theirs are slightly different from ours. Partially supported by NSF grant DMS0139813 and DMS0348258.     

Some Expansions in Basic Fourier Series and Related Topics

We consider explicit expansions of some elementary and q-functions in basic Fourier series introduced recently by Bustoz and Suslov. Natural q-extensions of the Bernoulli and Euler polynomials, numbers, and the Riemann zeta function are discussed as a by-product.     

A basis analog of theH-function of several variables

In the present paper we construct a basis analog of theH-function of several variables with the kernel depending on the products ofq-gamma functions, including, for example, theH-function and theG-function ofn variables. We obtain a sufficient condition for the convergence of the basis analog of theH-function ofn variables, integral representations, and contiguous relations. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 738–744, May, 2000.     

Theq-gamma function forx<0

F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!) _q =1(1+q)(1+q+q ²)...(1+q+q ²+...+q ^n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x –1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of _q asq changes and showed that asq1^–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of _q (x) forx<0 and allq>0. In addition to some basic properties of _q , we will study the behavior of the sequence {x _n (q)} of critical points asn orq changes.     

A sequence approach to linear perfect hash families

A linear (q ^d , q, t)-perfect hash family of size s in a vector space V of order q ^d over a field F of order q consists of a set of linear functionals from V to F with the following property: for all t subsets there exists such that is injective when restricted to F. A linear (q ^d , q, t)-perfect hash family of minimal size d(t − 1) is said to be optimal. In this paper, we extend the theory for linear perfect hash families based on sequences developed by Blackburn and Wild. We develop techniques which we use to construct new optimal linear (q ², q, 5)-perfect hash families and (q ⁴, q, 3)-perfect hash families. The sequence approach also explains a relationship between linear (q ³, q, 3)-perfect hash families and linear (q ², q, 4)-perfect hash families.      

Convexity andq-gamma function

Inequalities and convexity properties known for the gamma function are extended to theq-gamma function, 0<q<1. Applying some classical inequalities for convex functions, we deduce monotonicity results for several functions involving theq-gamma function. Further applications to the probability theory are given.     

q-Hardy-Berndt type sums associated with q-Genocchi type zeta and q-l-functions

The aim of this paper is to define new generating functions. By applying a derivative operator and the Mellin transformation to these generating functions, we define q-analogue of the Genocchi zeta function, q-analogue Hurwitz type Genocchi zeta function, and q-Genocchi type l-function. We define partial zeta function. By using this function, we construct p-adic interpolation functions which interpolate generalized q-Genocchi numbers at negative integers. We also define p-adic meromorphic functions on C_p. Furthermore, we construct new generating functions of q-Hardy-Berndt type sums and q-Hardy-Berndt type sums attached to Dirichlet character. We also give some new relations, related to these sums.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

Solomon's zeta functions over algebraic function fields

L. Solomon recently introduced a wide-ranging but concrete generalization of the Riemann and Dedkind zeta functions, as well as of Hey's zeta function for a simple algebra over the rationals. The coefficients of Solomon's zeta function give the numbers of certain types of sublattices in a given lattice over an order in a semisimple rational algebra. This paper studies the analogous zeta function and coefficients which arise for an order in a semi-simpleF _q (X) -algebra, whereF _q (X) is a field of rational functions over a finite fieldF _q . Use is made of the analogues for function fields of results on his zeta functions which were first conjectured by Solomon, and later established by C J Bushnell and l Reiner.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

Theq-gamma function forq>1

F. H. Jackson defined a generalization of the factorial function by $$1(1 + q)(1 + q + q^2 ) \cdot \cdot \cdot (1 + q + q^2 + \cdot \cdot \cdot + q^{n - 1} ) = (n!)_q $$ forq>0. He also generalized the gamma function, both for 0<q<1, and forq>1. Askey then obtained analogues of many of the classical facts about theq-gamma function for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=[(q ^x ?1)/(q?1)]f(x) is theq-gamma function. He also considered the behavior of theq-gamma function asq changes, and showed that asq→1^?, theq-gamma function becomes the ordinary gamma function. In this paper we will state two analogues of the Bohr-Mollerup theorem forq>1. It turns out that the log convexity off together with the initial condition and the functional equation no longer forcesf to be theq-gamma function. A stronger condition is needed than the log convexity, and two sufficient conditions are given in this paper. Also we will consider the behavior of theq-gamma function asq-changes forq>1.     

On relations for the multiple <Emphasis Type="Italic">q</Emphasis>-zeta values

We prove a new relation for the multiple q-zeta values (MqZV’s). It is a q-analogue of the Ohno-Zagier relation for the multiple zeta values (MZV’s). We discuss the problem of determining the dimension of the space spanned by MqZV’s over ℚ, and present an application to MZV. The first author is supported by Grant-in-Aid for Young Scientists (B) No. 17740026 and the second author is supported by Grant-in-Aid for Young Scientists (B) No. 17740089.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

Onq-difference equations andZ n decompositions of exp q function

Theq-extended hyperbolic functions ofn-th order {h _q,s(z)}s∈ Z _n which areZ _n-components of expq function form the set fundamental solutions of a simpleq-difference equation. Against the background ofq-deformed hyperbolic functions ofn-th order other natural extensions and related topics are considered. Apart from easy general solution of homogenous ordinaryq-difference equations ofn-th order main trigonometric-like identity known for hyperbolic functions ofn-th order is given itsq-commutative counterpart. Hint how to arrive at other identities is implicit in theq-treatment proposed. The paper constitutes an example of the application of the method of projections presented in Advances in Applied Clifford Algebras publication [19]; see also references to Ben Cheikh’s papers.     

On the admissible families of components of hamming codes

In this paper, the properties of the i-components of Hamming codes are described. We suggest constructions of the admissible families of components of Hamming codes. Each q-ary code of length m and minimum distance 5 (for q = 3, the minimum distance is 3) is shown to embed in a q-ary 1-perfect code of length n = (q ^m − 1)/(q − 1). Moreover, each binary code of length m+k and minimum distance 3k + 3 embeds in a binary 1-perfect code of length n = 2 ^m − 1.     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

On generalized Clifford algebraC 4 (n) andGL q (2;C) quantum group

The non commuting matrix elements of matrices from quantum groupGL _q (2;C) withq≡ω being then-th root of unity are given a representation as operators in Hilbert space with help ofC ₄ ⁽ⁿ⁾ generalized Clifford algebra generators appropriately tensored with unit 2×2 matrix infinitely many times. Specific properties of such a representation are presented. Relevance of generalized Pauli algebra to azimuthal quantization of angular momentum alà Lévy-Leblond [10] and to polar decomposition ofSU _q (2;C) quantum algebra alà Chaichian and Ellinas [6] is also commented. The case ofq∈C, |q|=1 may be treated parallely.     

‐differential equations for ‐classical polynomials and ‐Jacobi–Stirling numbers

We introduce, characterise and provide a combinatorial interpretation for the so‐called q‐Jacobi–Stirling numbers. This study is motivated by their key role in the (reciprocal) expansion of any power of a second order q‐differential operator having the q‐classical polynomials as eigenfunctions in terms of other even order operators, which we explicitly construct in this work. The results here obtained can be viewed as the q‐version of those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a q‐version of the Jacobi–Stirling numbers given by Gelineau and the second author.