Applications of the classical umbral calculus

We describe applications of the classical umbral calculus to bilinear generating functions for polynomial sequences, identities for Bernoulli and related numbers, and Kummer congruences.Dedicated to the Memory of Gian-Carlo Rota     

Restricted 123-avoiding Baxter permutations and the Padovan numbers

Baxter studied a particular class of permutations by considering fixed points of the composite of commuting functions. This class is called Baxter permutations. In this paper we investigate the number of 123-avoiding Baxter permutations of length n that also avoid (or contain a prescribed number of occurrences of) another certain pattern of length k. In several interesting cases the generating function depends only on k and is expressed via the generating function for the Padovan numbers.     

The generalized order-k Fibonacci–Pell sequence by matrix methods

In this paper, we consider the usual and generalized order-k Fibonacci and Pell recurrences, then we define a new recurrence, which we call generalized order-k F–P sequence. Also we present a systematic investigation of the generalized order-k F–P sequence. We give the generalized Binet formula, some identities and an explicit formula for sums of the generalized order-k F–P sequence by matrix methods. Further, we give the generating function and combinatorial representations of these numbers. Also we present an algorithm for computing the sums of the generalized order-k Pell numbers, as well as the Pell numbers themselves.     

q-Analogs of some congruences involving Catalan numbers

We present some variations on the Greene–Krammer?s identity which involve q-Catalan numbers. Our method reveals an intriguing analogy between these new identities and some congruences modulo a prime.     

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.     

The generalized Fibonomial matrix

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k+1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples.     

Restricted even permutations and Chebyshev polynomials

We study generating functions for the number of even (odd) permutations on n letters avoiding 132 and an arbitrary permutation τ on k letters, or containing τ exactly once. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind.     

q-Pell sequences and two identities of V.A. Lebesgue

We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.     

A summation on Bernoulli numbers

In this paper, our aim is to investigate the summation form of Bernoulli numbers B_n, such as . We derive some basic identities among them. These numbers can form a Seidel matrix. The upper diagonal elements of this Seidel matrix are called “the median Bernoulli numbers”. We determine the prime divisors of their numerators and denominators. And we characterize their ordinary generating function as the unique solution of some functional equation. At last, we also obtain the continued fraction representation of their ordinary generating function and their value of Hankel determinant.     

Matrices related to the Bell polynomials

In this paper, we study the matrices related to the partial exponential Bell polynomials and those related to the Bell polynomials with respect to Ω. As a result, the factorizations of these matrices are obtained, which give unified approaches to the factorizations of many lower triangular matrices. Moreover, some combinatorial identities are also derived from the corresponding matrix representations.     

Generalizing inplace multiplicity identities for integer compositions

In a recent paper, the authors gave two new identities for compositions, or ordered partitions, of integers. These identities were based on closely-related integer partition functions which have recently been studied. In the process, we also extensively generalized both of these identities. Since then, we asked whether one could generalize one of these results even further by considering compositions in which certain parts could come from t kinds (rather than just two kinds, which was the crux of the original result). In this paper, we provide such a generalization. A straightforward bijective proof is given and generating functions are provided for each of the types of compositions which arise. We close by briefly mentioning some arithmetic properties satisfied by the functions which count such compositions.     

Bernoulli matrix and its algebraic properties

In this paper, we define the generalized Bernoulli polynomial matrix B^(α)(x) and the Bernoulli matrix B. Using some properties of Bernoulli polynomials and numbers, a product formula of B^(α)(x) and the inverse of B were given. It is shown that not only B(x)=P[x]B, where P[x] is the generalized Pascal matrix, but also B(x)=FM(x)=N(x)F, where F is the Fibonacci matrix, M(x) and N(x) are the (n+1)×(n+1) lower triangular matrices whose (i,j)-entries are and , respectively. From these formulas, several interesting identities involving the Fibonacci numbers and the Bernoulli polynomials and numbers are obtained. The relationships are established about Bernoulli, Fibonacci and Vandermonde matrices.     

K. Saito's Conjecture for nonnegative eta products and analogous results for other infinite products

We prove that the Fourier coefficients of a certain general eta product considered by K. Saito are nonnegative. The proof is elementary and depends on a multidimensional theta function identity. The z=1 case is an identity for the generating function for p-cores due to Klyachko [A.A. Klyachko, Modular forms and representations of symmetric groups, J. Soviet Math. 26 (1984) 1879-1887] and Garvan, Kim and Stanton [F. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101 (1990) 1-17]. A number of other infinite products are shown to have nonnegative coefficients. In the process a new generalization of the quintuple product identity is derived.     

On stronger conjectures that imply the Erd?s-Moser conjecture

The Erd?s-Moser conjecture states that the Diophantine equation Sk(m)=mk, where Sk(m)=k¹+k²+?+k(m−1), has no solution for positive integers k and m with k?2. We show that stronger conjectures about consecutive values of the function Sk, that seem to be more naturally, imply the Erd?s-Moser conjecture.     

Symmetric identities on Bernoulli polynomials

In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and Bernoulli numbers, including the Miki identity.     

Some identities involving the binomial sequences

Using the exponential generating function and the Bell polynomials, we obtain several new identities for the binomial sequences. As applications, some interesting identities are established for the Abel polynomials, exponential polynomials and factorial powers.     

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers n₁,…,nm, nm+1=n₁, and any nonnegative integer r, the expression

     

Generalizations of two identities of Guo and Yang

In this paper, we provide generalizations of two identities of Guo and Yang [2] for the q-binomial coe?cients. This approach allows us to derive new convolution identities for the complete and elementary symmetric functions. New identities involving q-binomial coe?cients are obtained as very special cases of these results. A new relationship between restricted partitions and restricted partitions into parts of two kinds is derived in this context.     

A combinatorial identity with application to Catalan numbers

By a very simple argument, we prove that if l,m,n∈{0,1,2,…} then