in this paper we demonstrate the existence of a set of polynomials pi , 1 i  n , which arepositive semi-definite on an interval [a , b] and satisfy, partially, the conditions of polynomials found in thelagrange interpolation process. in other words, if a  a1  an  b is a given finite sequence of realnumbers, then pi (a j )  ij (ij is the kronecker delta symbol ) ; moreover, the sum of ...

We prove conjectures of the third author [L. Tevlin, Proc. FPSAC’07, Tianjin] on two new bases of noncommutative symmetric functions: the transition matrices from the ribbon basis have nonnegative integral coefficients. This is done by means of two composition-valued statistics on permutations and packed words, which generalize the combinatorics of Genocchi numbers.

In this paper, we introduce a new generalization of the Hermite polynomials via (p; q)-exponential generating function and investigate several properties and relations for mentioned polynomials including derivative property, explicit formula, recurrence relation, integral representation. We also de…ne a (p; q)analogue of the Bernstein polynomials and acquire their some formulas. We then provide...

By means of the generating function approach, we derive several summation formulae involving multiple harmonic numbers Hn,? (?), as well other combinatorial named after Bernoulli, Euler, Bell, Genocchi and Stirling.

The number of self-dual cyclic codes of length pk over GR(p2,m) is determined by the nullity of a certain matrix M(pk, i1). With the aid of Genocchi numbers, we determine the nullity of M(pk, i1) and hence determine completely the number of such codes.

In this paper we refine the well-known permutation statistic “descent” by fixing parity of (exactly) one of the descent’s numbers. We provide explicit formulas for the distribution of these (four) new statistics. We use certain differential operators to obtain the formulas. Moreover, we discuss connection of our new statistics to the Genocchi numbers. We also provide bijective proofs of some of...

In this paper, we study the properties of the generalized harmonic numbersHn,k,r(α, β). In particular, by means of the method of coefficients, generating functions and Riordan arrays, we establish some identities involving the numbers Hn,k,r(α, β), Cauchy numbers, generalized Stirling numbers, Genocchi numbers and higher order Bernoulli numbers. Furthermore, we obtain the asymptotic values of s...