Some results on sums of products of Bernoulli polynomials and Euler polynomials

In this paper, by the methods of partial fraction decomposition and generating functions, we establish an explicit expression for sums of products of l Bernoulli polynomials and n?l Euler polynomials, i.e., for sums $$S_n^{(k)}(y;l,k-l):= \sum_{\substack{j_1+\cdots+j_k=n\\j_1,\dots,j_k\geq0}} \binom {n}{j_1,\dots,j_k} B_{j_1}(x_1)\cdots B_{j_l}(x_l)E_{j_{l+1}}(x_{l+1}) \cdots E_{j_k}(x_k). $$ This result is then used to deal with various other types of sums of products of Bernoulli polynomials and Euler polynomials. Some of them are expressed in terms of $S_{n}^{(k)}(y;l,k-l)$ and can be computed directly, while the others satisfy certain recurrences and can be determined recursively. As a consequence, many known results are special cases of ours.     

Degenerate Bernoulli polynomials, generalized factorial sums, and their applications

We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.     

Explicit merit factor formulae for Fekete and Turyn polynomials

We give explicit formulas for the norm (or equivalently for the merit factors) of various sequences of polynomials related to the Fekete polynomials

where is the Legendre symbol. For example for an odd prime,

where is the class number of . Similar explicit formulas are given for various polynomials including an example of Turyn's that is constructed by cyclically permuting the first quarter of the coefficients of . This is the sequence that has the largest known asymptotic merit factor. Explicitly,

where denotes the nearest integer, satisfies

where

Indeed we derive a closed form for the norm of all shifted Fekete polynomials

Namely

and if .

     

Distribution of harmonic sums and Bernoulli polynomials modulo a prime

For a fixed integer s≥1, we estimate exponential sums with harmonic sums individually and on average, where H_s(n) is computed modulo a prime p. These bounds are used to derive new results about various congruences modulo p involving H_s(n). For example, our estimates imply that for any ɛ>0, the set {H_s(n):n<p^1/2+ɛ} is uniformly distributed modulo a sufficiently large p. We also show that every residue class λ can be represented as with max{n_ν|ν=1,. . . , 7}≤p^11/12+ɛ, and we obtain an asymptotic formula for the number of such representations. The same results hold also for the values B_p−r(n) of Bernoulli polynomials where r is fixed, complementing some results of W. L. Fouche. During the preparation of this paper, F. L. was supported in part by grants SEP-CONACYT 37259-E and 37260-E, and I. S. was supported in part by ARC grant DP0211459.     

Some identities on the Bernoulli, Euler and Genocchi polynomials via power sums and alternate power sums

In this paper, by the generating function method, we establish various identities concerning the (higher order) Bernoulli polynomials, the (higher order) Euler polynomials, the Genocchi polynomials and the degenerate higher order Bernoulli polynomials. Particularly, some of these identities are also related to the power sums and alternate power sums. It can be found that, many well known results, especially the multiplication theorems, and some symmetric identities demonstrated recently, are special cases of our results.     

A note on sums of products of Bernoulli numbers

In this work we obtain a new approach to closed expressions for sums of products of Bernoulli numbers by using the relation of values at non-positive integers of the important representation of the multiple Hurwitz zeta function in terms of the Hurwitz zeta function.     

Identities for Bernoulli polynomials and Bernoulli numbers

We prove that if m and \({\nu}\) are integers with \({0 \leq \nu \leq m}\) and x is a real number, then

1. $$\sum_{k=0 \atop k+m \, \, odd}^{m-1} {m \choose k}{k+m \choose \nu} B_{k+m-\nu}(x) = \frac{1}{2} \sum_{j=0}^m (-1)^{j+m} {m \choose j}{j+m-1 \choose \nu} (j+m) x^{j+m-\nu-1},$$ where B _n (x) denotes the Bernoulli polynomial of degree n. An application of (1) leads to new identities for Bernoulli numbers B _n . Among others, we obtain
2. $$\sum_{k=0 \atop k+m \, \, odd}^{m -1} {m \choose k}{k+m \choose \nu} {k+m-\nu \choose j}B_{k+m-\nu-j} =0 \quad{(0 \leq j \leq m-2-\nu)}. $$ This formula extends two results obtained by Kaneko and Chen-Sun, who proved (2) for the special cases j = 1, \({\nu=0}\) and j = 3, \({\nu=0}\) , respectively.

     

Transformation formulae for Dirichlet polynomials

Formulae of Voronoi-Atkinson type are proved for Dirichlet polynomials related to the Dirichlet series ζ²(s) = Σd(n)n^?s or ?(s) = Σa(n)n^?s, where the a(n) are the Fourier coefficients of a cusp form, a typical example being a(n) = τ(n), the Ramanujan function. Applications are given to a formula of Atkinson (Acta Math.81 (1949), 353–376) for the mean square of $\text{|ζ(}\text{1}\text{2}\text{+ it)|}$ and to the differences between consecutive zeros of ?(s) on the critical line in the case when all the a(n) are real.     

Summation formulae for finite cotangent sums

Recently, a half-dozen remarkably general families of the finite trigonometric sums were summed in closed-form by choosing a particularly convenient integration contour and making use of the calculus of residues. In this sequel, we show that this procedure can be further extended and we find the summation formulae, in terms of the higher order Bernoulli polynomials and the ordinary Bernoulli polynomials, for four general families of the finite cotangent sums.     

Generalization of Bernoulli polynomials

The Bernoulli polynomials are generalized and some properties of the resulting generalizations are presented.     

Infinite sums for Fibonacci polynomials and Lucas polynomials

The Ramanujan Journal - In this paper we establish certain infinite sums involving many arithmetical functions and the Fibonacci polynomials or the Lucas polynomials. Several of the sums are given...     

Small divisors of Bernoulli sums

Let ?= {?_i,i ≥1} be a sequence of independent Bernoulli random variables (P{?_i = 0} = P{?_i = 1 } = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums B_n=?₁+...+?_n, n=1,2..... We obtain an asymptotic estimate for the probability P{P^-(B_n) > >} for >≤n^{e/log log n}, c a positive constant.     

Asymptotic formulae for generalized Freud polynomials

We establish the Mehler–Heine type formulae for orthonormal polynomials with respect to generalized Freud weights. Using this type of asymptotics, we can give estimates of the value at the origin of these polynomials and of all their derivatives as well as the asymptotic behavior of the corresponding zeros.     

Higher-order convolutions for Bernoulli and Euler polynomials

We prove convolution identities of arbitrary orders for Bernoulli and Euler polynomials, i.e., sums of products of a fixed but arbitrary number of these polynomials. They differ from the more usual convolutions found in the literature by not having multinomial coefficients as factors. This generalizes a special type of convolution identity for Bernoulli numbers which was first discovered by Yu. Matiyasevich.     

Bernoulli numbers and sums of powers

There have been many studies of Bernoulli numbers since Jakob Bernoulli first used the numbers to compute sums of powers, 1 ^p + 2 ^p + 3 ^p + ··· + n^p , where n is any natural number and p is any non-negative integer. By examining patterns of these sums for the first few powers and the relation between their coefficients and Bernoulli numbers, the author hypothesizes and proves a new recursive algorithm for computing Bernoulli numbers, sums of powers, as well as m-ford sums of powers, which enrich the existing literatures of Bernoulli numbers.     

Explicit formulae for L-values in positive characteristic

We focus on the generating series for the rational special values of Pellarin’s \(L\) -series in \(1 \le s \le 2(q-1)\) indeterminates, and using interpolation polynomials we prove a closed form formula relating this generating series to the Carlitz exponential, the Anderson–Thakur function, and the Anderson generating functions for the Carlitz module. We draw several corollaries, including explicit formulae and recursive relations for Pellarin’s \(L\) -series in the same range of \(s\) , and divisibility results on the numerators of the Bernoulli–Carlitz numbers by monic irreducibles of degrees one and two.