Cited By
View all- Cao J(2017)Homogeneous q-partial difference equations and some applicationsAdvances in Applied Mathematics10.1016/j.aam.2016.11.00184:C(47-72)Online publication date: 1-Mar-2017
q-Taylor theorems, polynomial expansions, and interpolation of entire functions
Pages 125 - 146
Abstract
We establish q-analogues of Taylor series expansions in special polynomial bases for functions analytic in bounded domains and for entire functions whose maximum modulus M(r;f) satisfies |ln M(r;f)|≤A ln2r. This solves the problem of constructing such entire functions from their values at [aqn + q-n/a]/2, for 0<q< 1. Our technique is constructive and gives an explicit representation of the sought entire function. Applications to q-series identities are given.
References
[1]
{1} G.E. Andrews, R.A. Askey, R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
[2]
{2} R.A. Askey, J.A. Wilson, Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Memoirs of the American Mathematical Society, 319. Providence RI, 1985.
[3]
{3} R.P. Boas, Entire Functions, Academic Press, New York, 1954.
[4]
{4} R.P. Boas, R.C. Buck, Polynomial Expansions of Analytic Functions, Springer, Berlin, 1964.
[5]
{5} S. Cooper, The Askey-Wilson operator and the 6φ5 summation formula, preprint, March 1996.
[6]
{6} G. Gasper, Elementary derivations of summation and transformation formulas for q-series, in: M.E.H. Ismail, D.R. Masson, M. Rahman (Eds.), Special Functions, q-Series and Related Topics, Fields Institute Communication, Vol. 14, American Mathematical Society, Providence, RI, 1997, pp. 55-70.
[7]
{7} G. Gasper, M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 1990.
[8]
{8} A.O. Gelfond, Calculus of Finite Differences, Translated from the Russian, Hindustan Publishing Corporation, Delhi, 1971.
[9]
{9} G.H. Hardy, Ramanujan: Twelve Lectures on Subjects Suggested by his Life and Work, Chelsea, New York, 1978.
[10]
{10} P. Hartman, Difference equations: disconjugacy, principal solutions, Green's functions, complete monotonicity, Trans. Amer. Math. Soc. 246 (1978) 1-30.
[11]
{11} M.E.H. Ismail, The Askey-Wilson operator and summation theorems, in: M. Ismail, M.Z. Nashed, A. Zayed, A. Ghaleb (Eds.), Mathematical Analysis, Wavelets, and Signal Processing, Contemporary Mathematics, Vol. 190, American Mathematical Society, Providence, RI, 1995, pp. 171-178.
[12]
{12} M.E.H. Ismail, D. Stanton, Applications of q-Taylor theorems, J. Comp. Appl. Math. 153 (2003) 259-272.
[13]
{13} M.E.H. Ismail, J. Wilson, Asymptotic and generating relations for the q-Jacobi and the 4φ3 polynomials, J. Approx. Theory 36 (1982) 43-54.
[14]
{14} M.E.H. Ismail, R. Zhang, Diagonalization of certain integral operators, Adv. Math. 109 (1994) 1-33.
[15]
{15} M.E.H. Ismail, R. Zhang, New proofs of some q-series results, in: M.E.H. Ismail, H. Koelink (Eds.), Kluwer, to appear.
[16]
{16} B.H. Makar, The theory of coordinates, a book manuscript, 1973.
[17]
{17} G. Polya, Über ganzwertige ganze Funktionen, Ren. Circ. Mat. Palermo 40 (1915) 1-16.
[18]
{18} E.D. Rainville, Special Functions, Chelsea, the Bronx, 1971.
[19]
{19} J.P. Ramis, About the growth of entire functions solutions of linear algebraic q-difference equations, Ann. Fac. Sc. Toulouse 1 (6) (1992) 53-94.
[20]
{20} R.V. Wallisser, On entire functions assuming integer values in a geometric sequence, in: Théorie des Nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 981-989.
[21]
{21} E.T. Whittaker, G.N. Watson, Modern Analysis, 4th Edition, Cambridge University Press, Cambridge, 1927.
[22]
{22} J.M. Whittaker, Interpolatory Function Theory, Cambridge Tracts in Mathematics and Mathematical Physics, No. 33, Cambridge, 1935.
[23]
{23} J.M. Whittaker, Sur les Séries de Base de Polynomes Quelconques, Gauthier-Villars, Paris, 1949.
Index Terms
- q-Taylor theorems, polynomial expansions, and interpolation of entire functions
Recommendations
Applications of q-Taylor theorems
We establish two new q-analogues of a Taylor series expansion for polynomials using special Askey-Wilson polynomial bases. Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization ...
Applications of q-Taylor theorems
Proceedings of the sixth international symposium on orthogonal polynomials, special functions and their applicationsWe establish two new q-analogues of a Taylor series expansion for polynomials using special Askey-Wilson polynomial bases. Combining these expansions with an earlier expansion theorem we derive inverse relations and evaluate certain linearization ...
q-Blossoming and Hermite-Padé approximants to the q-exponential function
We derive explicit expressions for the type II Hermite-Padé approximants to the q-exponential function using the q-blossom of its partial sums. In particular, this derivation leads to an interpretation of the Padé approximants as control points of a q-...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Publisher
Academic Press, Inc.
United States
Publication History
Published: 01 July 2003
Author Tags
Qualifiers
- Article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- View Citations1Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 24 Jan 2025
Other Metrics
Citations
Cited By
View all- Cao J(2017)Homogeneous q-partial difference equations and some applicationsAdvances in Applied Mathematics10.1016/j.aam.2016.11.00184:C(47-72)Online publication date: 1-Mar-2017