140 research outputs found
Using integrals of squares of certain real-valued special functions to prove that the P\'olya \Xi^*(z) function, the functions K_{iz}(a), a > 0, and some other entire functions have only real zeros
Analogous to the use of sums of squares of certain real-valued special
functions to prove the reality of the zeros of the Bessel functions J_\alpha(z)
when \alpha \ge -1, confluent hypergeometric functions {}_0F_1(c; z) when c > 0
or 0 > c > -1, Laguerre polynomials L_n^\alpha(z) when \alpha \ge -2, Jacobi
polynomials P_n^{(\alpha,\beta)}(z) when \alpha \ge -1 and \beta \ge -1, and
some other entire special functions considered in G. Gasper [Using sums of
squares to prove that certain entire functions have only real zeros, in Fourier
Analysis: Analytic and Geometric Aspects, W. O. Bray, P. S. Milojevi\'c and C.
V. Stanojevi\'c, eds., Marcel Dekker, Inc., 1994, 171--186.], integrals of
squares of certain real-valued special functions are used to prove the reality
of the zeros of the P\'olya \Xi^*(z) function, the K_{iz}(a) functions when a >
0, and some other entire functions.Comment: 8 page
Lecture notes for an introductory minicourse on q-series
These lecture notes were written for a mini-course that was designed to
introduce students and researchers to {\it -series,} which are also called
{\it basic hypergeometric series} because of the parameter that is used as
a base in series that are ``{\it over, above or beyond}'' the {\it geometric
series}. We start by considering -extensions (also called -analogues) of
the binomial theorem, the exponential and gamma functions, and of the beta
function and beta integral, and then progress on to the derivations of rather
general summation, transformation, and expansion formulas, integral
representations, and applications. Our main emphasis is on methods that can be
used to {\bf derive} formulas, rather than to just {\it verify} previously
derived formulas
Using sums of squares to prove that certain entire functions have only real zeros
It is shown how sums of squares of real valued functions can be used to give
new proofs of the reality of the zeros of the Bessel functions
when confluent hypergeometric functions {}_0F_1(c\/; z) when
or , Laguerre polynomials when
and Jacobi polynomials when and Besides yielding new inequalities for where
is one of these functions, the derived identities lead to inequalities for
and which
also give new proofs of the reality of the zeros
Elementary derivations of summations and transformation formulas for q-series
We present some elementary derivations of summation and transformation
formulas for q-series, which are different from, and in several cases simpler
or shorter than, those presented in the Gasper and Bahman [1990] "Basic
Hypergeometric Series" book (which we will refer to as BHS), the Bailey [1935]
and Slater [1966] books, and in some papers; thus providing deeper insights
into the theory of q-series. Our main emphasis is on methods that can be used
to derive formulas, rather than to just verify previously derived or
conjectured formulas. In section 5 this approach leads to the derivation of a
new family of summation formulas for very well poised basic hypergeometric
series _{6+2k}W_{5+2k}, k = 1,2,.... Several of the observations in this paper
were presented, along with related exercises, in the author's minicourse on
"q-Series" at the Fields Institute miniprogram on "Special functions, q-Series
and Related Topics," June 12-14, 1995
q-Analogues of Some Multivariable Biorthogonal Polynomials
A q-analogue a pair of multivariable biorthogonal polynomials found by
M.V.Tratnik in 1989 is derived. The weight function is a product of a
multivariable version of the integrand in the Askey-Roy integral and of the
Askey-Wilson weight function in a single variable. In addition, a
biorthogonality relation is derived for certain bivariate extensions of the
-Racah polynomials.Comment: 16 pages. To appear in Theory and Applications of Special Functions.
A volume dedicated to Mizan Rahman, M. E. H. Ismail and E. Koelink, eds.,
Dev. Math., Kluwer Acad. Publ., Dordrech
Some curious q-series expansions and beta integral evaluations
We deduce several curious q-series expansions by applying inverse relations
to certain identities for basic hypergeometric series. After rewriting some of
these expansions in terms of q-integrals, we obtain, in the limit q -> 1, some
curious beta-type integral evaluations which appear to be new.Comment: 14 pages, dedicated to Dick Aske
Summation, transformation, and expansion formulas for multibasic theta hypergeometric series
After reviewing some fundamental facts from the theory of theta
hypergeometric series we derive, using indefinite summation, several summation,
transformation, and expansion formulas for multibasic theta hypergeometric
series. Some of the identities presented here generalize corresponding formulas
given in Chapter 11 of the Gasper and Rahman book [Basic hypergeometric series,
2nd ed., Encyclopedia of Mathematics And Its Applications 96, Cambridge
University Press, Cambridge, 2004].Comment: 18 pages; for the proceedings of the workshop "Elliptic Integrable
Systems" (RIMS, Kyoto, November 8-11, 2004), special volume of the Rokko
Lectures in Mathematic
Some Systems of Multivariable Orthogonal q-Racah polynomials
In 1991 Tratnik derived two systems of multivariable orthogonal Racah
polynomials and considered their limit cases. q-Extensions of these systems are
derived, yielding systems of multivariable orthogonal q-Racah polynomials, from
which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk,
q-Meixner, and q-Charlier polynomials follow as special or limit cases.Comment: 15 page
The Sums of a Double Hypergeometric Series and of the First m+1 Terms of 3F2(a,b,c;(a+b+1)/2,2c;1) when c = -m is a Negative Integer
A summation formula is derived for the sum of the first m+1 terms of the
3F2(a,b,c;(a+b+1)/2,2c;1) series when c = -m is a negative integer. This
summation formula is used to derive a formula for the sum of a terminating
double hypergeometric series that arose in another project by one of us (C.D.)Comment: 7 pages, added general remarks, another derivation of one of the
formulas, and references to related results in basic hypergeometric serie
Errata, updates of the references, etc., for the book Basic Hypergeometric Series
Here are the latest errata, etc., to the Gasper and Rahman "Basic
Hypergeometric Series" book. Any additional errata will be added to the end of
the last list.Comment: Additional errata and material adde
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