Related papers: On Polynomial Identities for Recursive Sequences
We introduce poly-Bernoulli polynomials in two variables by using a generalization of Stirling numbers of the second kind that we studied in a previous work. We prove the bi-variate poly-Bernoulli polynomial version of some known results on…
For a polynomial P, we consider the sequence of iterated integrals of ln P(x). This sequence is expressed in terms of the zeros of P(x). In the special case of ln(1 + x^2), arithmetic properties of certain coefficients arising are…
We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their…
In the paper we provide some polynomial identities for finite-dimensional algebras. A list of well known single polynomial identities is exposed and the classification of all $2$-dimensional algebras with respect to these identities is…
In this paper, we introduce a novel identity for generalized Euler polynomials, leading to further generalizations for several relations involving classical Euler numbers, Euler polynomials, Genocchi polynomials, and Genocchi numbers.
We prove a conjecture that arose in the context of a subspace enumeration problem over finite fields. We prove, more generally, a bibasic, double-sum identity, which extends a $q$-analogue of the (terminating) binomial theorem.
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…
We propose and prove a new polynomial identity that implies Schur's partition theorem. We give combinatorial interpretations of some of our expressions in the spirit of Kur\c{s}ung\"oz. We also present some related polynomial and $q$-series…
Some applications of a result, which is proved recently, is considered. We first prove three determinantal identities concerning the binomial coefficient and Stirling numbers of the first and the second kind. We also easily obtain the…
In this paper we give describe a new connection between the dilogarithm function and solutions to Pell's equation $x^2-ny^2 = \pm 1$. For each solution $x,y$ to Pell's equation we obtain a dilogarithm identity whose terms are given by the…
We obtain similar types of conclusions as that of Br\"{u}ck [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover, a number of examples have been exhibited to justify the…
We consider sequences of the type $A_n=6A_{n-1}-A_{n-2}, A_0=r, A_1=s$ ($r$ and $s$ integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four…
We derive some Fibonacci and Lucas identities which contain inverse binomial coefficients. Extension of the results to the general Horadam sequence is possible, in some cases.
Power series in which the summand satisfies a linear recurrence relation with polynomial coefficients are shown to be the solution of a linear differential or algebraic equation. Solving the associated differential or algebraic equation…
We give a short proof of polynomial recurrence with large intersection for additive actions of finite-dimensional vector spaces over countable fields on probability spaces, improving upon the known size and structure of the set of strong…
The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of…
We consider the greatest common divisor (GCD) of all sums of $k$ consecutive terms of a sequence $(S_n)_{n\geq 0}$ where the terms $S_n$ come from exactly one of following six well-known sequences' terms: Pell $P_n$, associated Pell $Q_n$,…
We derive several symmetric identities for Bernoulli and Euler polynomials which imply some known identities. Our proofs depend on the new technique developed in part I and some identities obtained in [European J. Combin. 24(2003),…
In a recent letter, new representations were proposed for the pair of sequences ($\gamma,\delta$), as defined formally by Bailey in his famous lemma. Here we extend and prove this result, providing pairs ($\gamma,\delta$) labelled by the…
Utilizing spectral residues of parameterized, recursively defined sequences, we develop a general method for generating identities of composition sums. Specific results are obtained by focusing on coefficient sequences of solutions of first…