New identities involving Bernoulli and Euler polynomials

Using the finite difference calculus and differentiation, we obtain several new identities for Bernoulli and Euler polynomials; some extend Miki's and Matiyasevich's identities, while others generalize a symmetric relation observed by Woodcock and some results due to Sun.     

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.     

Congruences involving Bernoulli and Euler numbers

Let [x] be the integral part of x. Let p>5 be a prime. In the paper we mainly determine , , and in terms of Euler and Bernoulli numbers. For example, we have

     

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.     

Congruences involving Bernoulli polynomials

Let {B_n(x)} be the Bernoulli polynomials. In the paper we establish some congruences for , where p is an odd prime and x is a rational p-integer. Such congruences are concerned with the properties of p-regular functions, the congruences for and the sum , where h(d) is the class number of the quadratic field of discriminant d and p-regular functions are those functions f such that are rational p-integers and for n=1,2,3,… . We also establish many congruences for Euler numbers.     

A geometric perspective on counting non-negative integer solutions and combinatorial identities

We consider the effect of constraints on the number of non-negative integer solutions of x+y+z = n, relating the number of solutions to linear combinations of triangular numbers. Our approach is geometric and may be viewed as an introduction to proofs without words. We use this geometrical perspective to prove identities by counting the number of solutions in two different ways, thereby combining combinatorial proofs and proofs without words.     

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.     

Simple arguments on consecutive power residues

By some extremely simple arguments, we point out the following:

(i)

If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.

(ii)

Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .

     

Restricted colored permutations and Chebyshev polynomials

Several authors have examined connections between restricted permutations and Chebyshev polynomials of the second kind. In this paper we prove analogues of these results for colored permutations. First we define a distinguished set of length two and length three patterns, which contains only 312 when just one color is used. Then we give a recursive procedure for computing the generating function for the colored permutations which avoid this distinguished set and any set of additional patterns, which we use to find a new set of signed permutations counted by the Catalan numbers and a new set of signed permutations counted by the large Schröder numbers. We go on to use this result to compute the generating functions for colored permutations which avoid our distinguished set and any layered permutation with three or fewer layers. We express these generating functions in terms of Chebyshev polynomials of the second kind and we show that they are special cases of generating functions for involutions which avoid 3412 and a layered permutation.     

On Euler numbers modulo powers of two

To determine Euler numbers modulo powers of two seems to be a difficult task. In this paper we achieve this and apply the explicit congruence to give a new proof of a classical result due to M.A. Stern.     

On Delannoy numbers and Schröder numbers

The nth Delannoy number and the nth Schröder number given by

     

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.     

On sums of binomial coefficients and their applications

In this paper we study recurrences concerning the combinatorial sum and the alternate sum , where m>0, n?0 and r are integers. For example, we show that if n?m-1 then

     

On some identities for the Fibonomial coefficients via generating function

Some new identities for the Fibonomial coefficients are derived. These identities are related to the generating function of the kth powers of the Fibonacci numbers. Proofs are based on manipulation with the generating function of the sequence of “signed Fibonomial triangle”.     

Note on some congruences of Lehmer

In the paper, we generalize some congruences of Lehmer and prove that for any positive integer n with (n,6)=1

     

On some polynomials enumerating Fully Packed Loop configurations: Evaluation at negative values

In this article, we are interested in the enumeration of Fully Packed Loop configurations on a grid with a given noncrossing matching. These quantities also appear as the groundstate components of the O(n)$O(n)$ Loop model as conjectured by Razumov and Stroganov and recently proved by Cantini and Sportiello.     

Symmetric polynomials, Pascal matrices, and Stirling matrices

We consider lower-triangular matrices consisting of symmetric polynomials, and we show how to factorize and invert them. Since binomial coefficients and Stirling numbers can be represented in terms of symmetric polynomials, these results contain factorizations and inverses of Pascal and Stirling matrices as special cases. This work generalizes that of several other authors on Pascal and Stirling matrices.     

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.     

Legendre inversions and balanced hypergeometric series identities

By means of Legendre inverse series relations, we prove two terminating balanced hypergeometric series formulae. Their reversals and linear combinations yield several known and new hypergeometric series identities.