The purpose of this work is to compute the successive derivatives of Fibonacci type polynomials in two variables; polynomials these introduced by G. Ozdemir,Y. Simsek in [3] and generalized by G. Ozdemir, Y. Simsek and G. Milovanović in [2] to a higher order. In addition we construct their recursive formula different of that given in Theorem 2.2 [3] p.6. Finally we define a novel generalized class of those polynomials similar to that given in [1] and found its recursive formula.
First, we prove explicit formula for the symmetric sum (,) + (,) which is a new reciprocity law for the sums above. This formula can be seen as a complement to the Bettin-Conrey result [13, Theorem 1]. Second, we establish an asymptotic formula for (,). Finally, by use of continued fraction theory, we give a formula for (,) in terms of continued fraction of .
Recently, several works are done on the generalized Dedekind‐Vasyunin sum
Capq=−qa∑k=1q−1ζ−a,kpqcotπkq,
where
a∈double-struckC,p and q are positive coprime integers, and ζ(a,x) denotes the Hurwitz zeta function. We prove explicit formula for the symmetric sum
Capq+Caqp,
which is a new reciprocity law for the sum
Ca()pq. Our result is a complement to recent results dealing with the sum
Capq−qp1+aCa−qp,
studied by Bettin‐Conrey and then by Auli‐Bayad‐Beck. Accidentally, when a = 0, our reciprocity formula improves the known result in a previous study.
The purpose of this work is to give a positive answer to two questions asked by professor Yilmaz Simsek in a recent paper [6] concerning special numbers B(n, k) for computing negative order Euler numbers.
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