Discrete generating functions. (English. Russian original) Zbl 1536.39010
Math. Notes 114, No. 6, 1087-1093 (2023); translation from Prikl. Mat. Fiz. 55, No. 2, 89-95 (2023).
Summary: The notion of a discrete generating function is defined. The definition uses the falling factorial instead of a power function. A functional equation for the discrete generating function of a solution to a linear difference equation with constant coefficients is found. For the discrete generating function of a solution to a linear difference equation with polynomial coefficients, the notion of D-finiteness is introduced and an analog of Stanley’s theorem is proved; namely, a condition for the D-finiteness of the discrete generating function of a solution to such an equation is obtained.
MSC:
| 39B22 | Functional equations for real functions |
| 39A06 | Linear difference equations |
| 39A70 | Difference operators |
| 33C20 | Generalized hypergeometric series, \({}_pF_q\) |
Keywords:
generating function; finiteness; recursiveness; generating series; forward difference operatorReferences:
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