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Generating functions for B-splines with knots in geometric or affine progression. (English) Zbl 1316.65020
The authors obtain generating functions for B-splines. Since there is no structure on the knots when they are completely arbitrary, any interesting property about the corresponding B-splines cannot be derived. Here explicit formulas for the generating functions of B-splines with knots sequences $$\{x_k\}$$ in affine progression $$x_{k+1}=qx_k+h$$ are given, with $$q$$ and $$h$$ fixed parameters, and in the particular case ($$h=0$$) of geometric progression they are considered in terms of $$q$$-exponential functions.
To find an explicit formula for the generating functions by the de Boor recurrence over knots in affine progression and the definition of $$q$$-derivative, a partial differential equation is constructed, whose solution is the generating function for B-splines in affine progression.
Such generating functions are then used to obtain both known and new identities, as a generalization of the Schoenberg identity and an explicit expression for the moments of these B-splines.
Finally, this discussion includes two interesting special cases, i.e., uniform B-splines with knots at the integers ($$q=h=1$$) and nonuniform B-splines with knots at the $$q$$-integers ($$q\not=1$$, $$h=1$$).

##### MSC:
 65D07 Numerical computation using splines 05A30 $$q$$-calculus and related topics 05A15 Exact enumeration problems, generating functions 41A15 Spline approximation
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