On Viète-like formulas

New formulas for Stirling-like numbers and Dobinski-like formulas

Extensions of the Stirling numbers of the second kind and Dobinski-like formulas are proposed in a series of exercises for graduetes. Some of these new formulas recently discovered by me are to be found in A.K.Kwaśniewski’s source paper [1]. These extensions naturally encompass the well known q-extensions.The indicatory references are to point at a part of the vast domain of the foundations of ...

A complete view of Viète-like infinite products with Fibonacci and Lucas numbers

On umbral extensions of Stirling numbers and Dobinski-like formulas

ψ-umbral extensions of the Stirling numbers of the second kind are considered and the resulting new type of Dobinski-like formulas are discovered. These extensions naturally encompass the two well known q-extensions .The further consecutive ψ-umbral extensions of CarlitzGould q-Stirling numbers are therefore realized here in a two-fold way . The fact that the umbral q-extended Dobinski formula ...

Some q-Dixon-like summation formulas

We give a q-analogue of some Dixon-like summation formulas obtained by Gould and Quaintance [Fibonacci Quart. 48 (2010), 56–61] and Chu [Integral Transforms Spec. Funct. 23 (2012), 251–261], respectively. For example, we prove that

On Birkhoff Quadrature Formulas

In an earlier work the author has obtained new quadrature formulas (see (1.3)) based on function values and second derivatives on the zeros of nn(i) as defined by (1.2). The proof given earlier was quite long. The object of this paper is to provide a proof of this quadrature formula which is extremely simple and indeed does not even require the use of fundamental polynomials of (0,2) interpolat...