Some finite congruence lattices, I
Some finite generalizations of Euler's pentagonal number theorem
Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem.
Some formulas for the central trinomial and Motzkin numbers.
Some formulas that enumerate certain partitions and graphs
Some further results on Legendre numbers.
Some general classes of comatching graphs.
Some general problems on the number of parts in partitions
Some generalized Fibonacci polynomials.
Some generating functions for polynomials
Some geometric probability problems involving the Eulerian numbers.
Some identities for Chebyshev polynomials.
Some identities for factorials and binomial coefficients
Some inequalities for Kurepa's functions.
Some inequalities in functional analysis, combinatorics, and probability theory.
Some interesting series arising from the power series expansion of .
Some interpretations of the -Fibonacci numbers
In this paper we consider two parameters generalization of the Fibonacci numbers and Pell numbers, named as the -Fibonacci numbers. We give some new interpretations of these numbers. Moreover using these interpretations we prove some identities for the -Fibonacci numbers.
Some new facts about group 𝒢 generated by the family of convergent permutations
The aim of this paper is to present some new and essential facts about group 𝒢 generated by the family of convergent permutations, i.e. the permutations on ℕ preserving the convergence of series of real terms. We prove that there exist permutations preserving the sum of series which do not belong to 𝒢. Additionally, we show that there exists a family G (possessing the cardinality equal to continuum) of groups of permutations on ℕ such that each one of these groups is different than 𝒢 and is composed...
Some new infinite families of congruences modulo 3 for overpartitions into odd parts
Let denote the number of overpartitions of n in which only odd parts are used. Some congruences modulo 3 and powers of 2 for the function have been derived by Hirschhorn and Sellers, and Lovejoy and Osburn. In this paper, employing 2-dissections of certain quotients of theta functions due to Ramanujan, we prove some new infinite families of Ramanujan-type congruences for modulo 3. For example, we prove that for n, α ≥ 0, .
Some new transformations for Bailey pairs and WP-Bailey pairs
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding transformations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
Some observations and solutions to short and long global Nim.