Set Of Hypergeometric Functions Research Articles

Quaternionic modular forms and exceptional sets of hypergeometric functions

We determine the exceptional sets of hypergeometric functions corresponding to the (2, 4, 6) triangle group by relating them to values of certain quaternionic modular forms at CM points. We prove a result on the number fields generated by exceptional values, and by using modular polynomials we explicitly compute some examples.

THE ACTION OF HECKE OPERATORS ON HYPERGEOMETRIC FUNCTIONS

AbstractWe study the action of the Hecke operators Un on the set of hypergeometric functions, as well as on formal power series. We show that the spectrum of these operators on the set of hypergeometric functions is the set {na:n∈ℕ,a∈ℤ}, and that the polylogarithms play an important role in the study of the eigenfunctions of the Hecke operators Un on the set of hypergeometric functions. As a corollary of our results on simultaneous eigenfunctions, we also obtain an apparently unrelated result regarding the behavior of completely multiplicative hypergeometric coefficients.

On the Sign of a Set of Hypergeometric Functions (Richard Askey)

On the Sign of a Set of Hypergeometric Functions (Richard Askey)

On the Sign of a Set of Hypergeometric Functions

On the Sign of a Set of Hypergeometric Functions

Aberration expansion and evaluation of the quasi-gaussian beam by a set of orthogonal functions

A set of orthogonal functions, convenient for the aberration analysis of the quasi-gaussian beams such as the laser TEM00 mode is derived. It is a set of hypergeometric functions. We can expand the aberration function of a quasi-gaussian beam in terms of it, and can calculate the diffraction patterns of aberrations. This set of functions has some characteristics of orthogonality to each other and relations to the Bessel functions. It bears a resemblance to the Zernike’s circle polynomials, but has some different characteristics. It is useful in holography, optical communications, and optical measurement.