Generating functions for B-Splines with knots in geometric or affine progression

Dişibüyük Ç., Budakçı G., Goldman R., Oruç H.

CALCOLO, vol.51, no.4, pp.599-613, 2014 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 51 Issue: 4
  • Publication Date: 2014
  • Doi Number: 10.1007/s10092-013-0102-8
  • Journal Name: CALCOLO
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.599-613
  • Keywords: B-splines, Generating functions, Knots in geometric progression, Knots in affine progression, q-Derivatives, q-Exponentials
  • Dokuz Eylül University Affiliated: Yes


We derive explicit formulas for the generating functions of B-splines with knots in either geometric or affine progression. To find generating functions for B-splines whose knots have geometric or affine ratio q, we construct a PDE for these generating functions in which classical derivatives are replaced by q-derivatives. We then solve this PDE for the generating functions using q-exponential functions. We apply our generating functions to derive some known and some novel identities for B-splines with knots in geometric or affine progression, including a generalization of the Schoenberg identity, formulas for sums and alternating sums, and an explicit expression for the moments of these B-splines. Special cases include both the uniform B-splines with knots at the integers and the nonuniform B-splines with knots at the q-integers.