An !-sequence is defined by an = !an−1 − an−2, with initial conditions a0 = 0, a1 = 1. These !-sequences play a remarkable role in partition theory, allowing !generalizations of the Lecture Hall Theorem and Euler’s Partition Theorem. These special properties are not shared with other sequences, such as the Fibonacci sequence, defined by second-order linear recurrences. The !-sequence gives rise...

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials derive interesting relations. We also obtain new properties by making use of power series expansions q-trigonometric functions, q-exponential q-analogues the binomial theorem. By using Mathematica program, computational formulae graphical representation for aforementioned are obtained. a partial derivative o...

This paper proves an identity between flagged Schur polynomials, giving a duality row flags and column flags. generalises both the binomial determinant theorem due to Gessel Viennot symmetric function Aitken. As corollaries we obtain lifts of \(q\)-binomial coefficients polynomials. Our method is path counting argument on novel lattice generalising that used by Viennot.

This paper considers a variety of parity questions connected with classical partition identities of Euler, Rogers, Ramanujan and Gordon. We begin by restricting the partitions in the Rogers-Ramanujan-Gordon identities to those wherein even parts appear an even number of times. We then take up questions involving sequences of alternating parity in the parts of partitions. This latter study leads...

The author uses an elementary lemma on primes dividing binomial coefficients and estimates for primes in arithmetic progressions to sharpen a theorem of J. Rickert on simultaneous approximation to pairs of algebraic numbers. In particular, it is proven that max {∣∣∣∣√2− p1 q ∣∣∣∣ , ∣∣∣∣√3− p2 q ∣∣∣∣} > 10−10q−1.8161 for p1, p2 and q integral. Applications of these estimates are briefly discussed.

For a positive composite integer n, we investigate the residues of ( mn k ) for positive integers m and k. First, we discuss divisibility of such coefficients. Then we study congruence identities relating products of binomial coefficients modulo n. Certainly the Chinese Remainder Theorem can be used in combination with prime power results to evaluate binomial coefficients modulo a composite. Ho...

A new type of sl3 basic hypergeometric series based on Macdonald polynomials is introduced. Besides a pair of Macdonald polynomials attached to two different sets of variables, a key-ingredient in the sl3 basic hypergeometric series is a bisymmetric function related to Macdonald’s commuting family of q-difference operators, to the sl3 Selberg integrals of Tarasov and Varchenko, and to alternati...

I present a new algorithm for computing binomial coefficients modulo 2N. The proposed method has an O(N3·Multiplication(N)+N4) preprocessing time, after which a binomial coefficient C(P, Q) with 0≤Q≤P≤2N-1 can be computed modulo 2N in O(N2·log(N)·Multiplication(N)) time. Multiplication(N) denotes the time complexity of multiplying two N-bit numbers, which can range from O(N2) to O(N·log(N)·log(...