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Séminaire Lotharingien de Combinatoire, B76c (2017), 39 pp.

# Volker Strehl

# Lacunary Laguerre Series
from a Combinatorial Perspective

**Abstract.**
In recent work,
Babusci, Dattoli, G\'orska, and Penson
have presented a number of
lacunary generating functions
for the generalized Laguerre
polynomials *L*_{n}^{(\alpha)}(*x*),
i.e., series of the type
*\sum*_{n >= 0}
*c*_{n} L_{2n}^{(\alpha)}(*x*) *t*^{n},
by a method closely related to
umbral calculus.
This work is complemented here,
deriving many of their results by
interpreting Laguerre polynomials
combinatorially as enumerators for
discrete structures
(injective partial functions).
This combinatorial view pays in
that it suggests natural extensions
and gives a deeper insight into
the known formulas.

Received: December 21, 2016.
Accepted: February 25, 2017.
Final Version: March 22, 2017.

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