q-Taylor theorems, polynomial expansions, and interpolation of entire functions

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.

[1]  Mourad E. H. Ismail,et al.  Applications of q-Taylor theorems , 2003 .

[2]  F. H. Jackson.  q-Difference Equations , 1910 .

[3]  D. Lambie,et al.  Modern Analysis , 1966, Nature.

[4]  George E. Andrews,et al.  Special Functions: Partitions , 1999 .

[5]  G. Pólya,et al.  Ueber ganzwertige ganze funktionen , 1915 .

[6]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[7]  J. Ramis,et al.  About the growth of entire functions solutions of linear algebraic $q$-difference equations , 1992 .

[8]  Rolf Wallisser.  On entire functions assuming integer values in a geometric sequence , 1989 .

[9]  Mourad E. H. Ismail,et al.  Asymptotic and generating relations for the q-Jacobi and 4φ3 polynomials , 1982 .

[10]  A. O. Gelʹfond.  Calculus of finite differences , 1973 .

[11]  Mourad E. H. Ismail,et al.  Diagonalization of Certain Integral Operators , 1994 .

[12]  R. Askey,et al.  Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials , 1985 .

[13]  Philip Hartman,et al.  Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity , 1978 .

[14]  J. M. Whittaker.  Interpolatory function theory , 1935 .

[15]  G. Hardy,et al.  Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work , 1978 .

[16]  George Gasper.  Elementary derivations of summations and transformation formulas for q-series , 1996 .

[17]  J. M. Whittaker,et al.  Sur les séries de base de polynomes quelconques , 1949 .

[18]  Ralph P. Boas,et al.  Polynomial Expansions of Analytic Functions. , 1959 .