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In this paper we show some identities come from the q -binomial theorem and the triple product of Jacobi. Some of these identities relating the function sum of divisor of a positive integer n and the number of integer partitions of n . As corollary we found the next equation, for n ≥ 1 . ∑ l = 1 n k l l ! ∑ ( w 1 , w 2 , … , w l ) ∈ C n σ 1 ( w 1 ) σ 1 ( w 2 ) ⋯ σ 1 ( w l ) w 1 w 2 ⋯ w l = p k ( n ) , where σ 1 ( n ) is the sum of all positive divisors of n , p k ( n ) is the number of k -colored integer partitions of n , and C n is the set of integer compositions of n .

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

We introduce two new subclasses of analytic functions in the open symmetric unit disc using a linear operator associated with the q-binomial theorem. In addition, we discuss inclusion relations and properties preserving integral operators for functions in these classes. This paper generalizes some known results, as well as provides some new ones.

In this paper, we consider some generalized Fibonomial sums formulae and then prove them by using the Cauchy binomial theorem and q— Zeilberger algorithm in Mathematica session.

In this article, we make use of the q-binomial theorem to introduce and study two new subclasses ℵ(αq,q) and ℵ(α,q) of meromorphic functions in the open unit disk U, that is, analytic functions in the punctured unit disk U[sup.∗] =U\\0=z:z∈C and 0<|z|<1. We derive inclusion relations and investigate an integral operator that preserves functions which belong to these function classes. In addition, we establish a strict inequality involving a certain linear convolution operator which we introduce in this article. Several special cases and corollaries of our main results are also considered.

Newton's Binomial Theorem applied at the rate of 2 with the formula: (a1+a2)n=∑r=0nC(n,r)a1n−ra2r Problems in algebra are not limited binomial. Binomial only is not enough, so that multinomial is necessary. Multinomial Theorem has the formula: (a1+a2+...+ak)n=∑n1,n2,...,nk≥0n!n1!n2!...nk!a1n1a2n2...aknk The use of theorem in binomial problem is less practical so that Pascal Triangle is prefered, for easier use Pascal's Triangle. Solution of Triangle with the theorem multinomial problem more complicated. By analyzing multinomial through binomial form, can be obtained from modification that allows the Pascal triangle. The focus of the discussion is to determine Pascal Modified Triangular shape of a multinomial. This basic research using descriptive method, by analyzing the relevant theory is based on the study of literature. The results obtained are Modified Pascal's Triangle, which facilitates the work in the multinomial.

We prove two inequalities for the Mittag-Leffler function, namely that the function log E α ( x α ) is sub-additive for 0 < α < 1 , and super-additive for α > 1 . These assertions follow from two new binomial inequalities, one of which is a converse to the neo-classical inequality. The proofs use a generalization of the binomial theorem due to Hara and Hino (Bull London Math Soc 2010). For 0 < α < 2 , we also show that E α ( x α ) is log-concave resp. log-convex, using analytic as well as probabilistic arguments.

By employing the generating function approach, 16 triple sums for circular binomial products of binomial coefficients are examined. Recurrence relations and generating functions are explicitly determined. These symmetric sums may find potential applications in the analysis of algorithms, symbolic calculus, and computations in theoretical physics.

We provide several new q-congruences for truncated basic hypergeometric series, mostly of arbitrary order. Our results include congruences modulo the square or the cube of a cyclotomic polynomial, and in some instances, parametric generalizations thereof. These are established by a variety of techniques including polynomial argument, creative microscoping (a method recently introduced by the first author in collaboration with Zudilin), Andrews’ multiseries generalization of the Watson transformation, and induction. We also give a number of related conjectures including congruences modulo the fourth power of a cyclotomic polynomial.