22 research outputs found
Irrationality proof of a -extension of using little -Jacobi polynomials
We show how one can use Hermite-Pad\'{e} approximation and little -Jacobi
polynomials to construct rational approximants for . These numbers
are -analogues of the well known . Here , with
an integer greater than one. These approximants are good enough to show the
irrationality of and they allow us to calculate an upper bound for
its measure of irrationality: . This is sharper than the upper bound given by Zudilin (\textit{On the
irrationality measure for a -analogue of }, Mat. Sb. \textbf{193}
(2002), no. 8, 49--70).Comment: 13 pages, one reference was corrected, two were adde
Multiple little q-Jacobi polynomials
We introduce two kinds of multiple little q-Jacobi polynomials by imposing
orthogonality conditions with respect to r discrete little q-Jacobi measures on
the exponential lattice q^k (k=0,1,2,3,...), where 0 < q < 1. We show that
these multiple little q-Jacobi polynomials have useful q-difference properties,
such as a Rodrigues formula (consisting of a product of r difference
operators). Some properties of the zeros of these polynomials and some
asymptotic properties will be given as well.Comment: 15 page
Discrete integrable systems generated by Hermite-Pad\'e approximants
We consider Hermite-Pad\'e approximants in the framework of discrete
integrable systems defined on the lattice . We show that the
concept of multiple orthogonality is intimately related to the Lax
representations for the entries of the nearest neighbor recurrence relations
and it thus gives rise to a discrete integrable system. We show that the
converse statement is also true. More precisely, given the discrete integrable
system in question there exists a perfect system of two functions, i.e., a
system for which the entire table of Hermite-Pad\'e approximants exists. In
addition, we give a few algorithms to find solutions of the discrete system.Comment: 20 page
Lambert series in analytic number theory
Annotated bibliography of 18th, 19th, and early 20th century works involving
Lambert series. A tour of 19th and early 20th century analytic number theory.Comment: 42 page