The Knop–Sahi interpolation Macdonald polynomials are inho-mogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polyno-mials to study a new type of basic hypergeometric series of type gl n. Our main results include a new q-binomial theorem, new q-Gauss sum, and several transformation formulae for gl n series.

In investigating the properties of a certain class of homogeneous polynomials, we discovered an identity satisfied by their coefficients which involves simple 2F1 Gauss hypergeometric functions. This result appears to be new and we supply a direct proof. The simplicity of the identity is suggestive of a deeper result.

An overpartition pair is a combinatorial object associated with the q-Gauss identity and the 1ψ1 summation. In this paper, we prove identities for certain restricted overpartition pairs using Andrews’ theory of q-difference equations for well-poised basic hypergeometric series and the theory of Bailey chains.

The Knop–Sahi interpolation Macdonald polynomials are inhomogeneous and nonsymmetric generalisations of the well-known Macdonald polynomials. In this paper we apply the interpolation Macdonald polynomials to study a new type of basic hypergeometric series of type gln. Our main results include a new q-binomial theorem, new q-Gauss sum, and several transformation formulae for gln series.

, where one can also find sample input and output. ∞ k=0 (a) k (b) k k!(c) k x k , (where (z) k := z(z + 1)(z + 2) · · · (z + k − 1)), that nowadays is more commonly denoted by 2 F 1 a, b c ; x , has a long and distinguished history, going back to Lehonard Euler and Carl Friedrich Gauss. It was also one of Ramanujan's favorites. Under the guise of binomial coefficient sums it goes even further ...

Some aspects of Aomoto’s generalized hypergeometric functions on Grassmannian spaces Gr(k+1, n+1) are reviewed. Particularly, their integral representations in terms of twisted homology and cohomology are clarified with an example of the Gr(2, 4) case which corresponds to Gauss’ hypergeometric functions. The cases of Gr(2, n + 1) in general lead to (n + 1)-point solutions of the Knizhnik-Zamolo...

Boltzmann relations are widely used in semiconductor physics to express the charge-carrier densities as a function of Fermi level and temperature. However, these simple exponential only apply sharp band edges conduction valence bands. In this article, we present generalization accounting for tails. To end, required Fermi-Dirac integral is first recast Gauss hypergeometric function, followed by ...

Finite-part integration is a recently introduced method of evaluating convergent integrals by means the finite-part divergent [E. A. Galapon, Proc. R. Soc., A 473, 20160567 (2017)]. Current application involves exact and asymptotic evaluation generalized Stieltjes transform ∫0af(x)/(ω+x)ρdx under assumption that extension f(x) in complex plane entire. In this paper, elaborated further extended ...

The exact analytical values of the position and momentum information entropies for the stationary states of the one-dimensional quantum harmonic oscillator are only known for the ground (n = 0) and rst excited (n = 1) states. In the general case, the problem of calculating these entropies reduces to the evaluation of the logarithmic potential V n (t) = ? R 1 ?1 (H n (x)) 2 ln jx ? tje ?x 2 dx a...

Building on the developments of many people including Evans, Greene, Katz, McCarthy, Ono, Roberts, and Rodriguez-Villegas, we consider period functions for hypergeometric type algebraic varieties over finite fields consequently study in a manner that is parallel to classical functions. Using comparison between gamma function its field analogue Gauss sum, give systematic way obtain certain types...