For an odd prime p, let Ep(X)=∑n=0∞anXn∈Fp[[X]] denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series G(Xp) ∈ Fp[ [ X] ] , such that Lp-1(-T(X))(X)=Ep(X)·G(Xp) , where T(X)=∑i=1∞Xpi and Lp-1(α)(X) denotes the (generalized) Laguerre polynomial of degree p- 1 . We prove that G(Xp)=∑n=0∞(-1)nanpXnp , and show that it satisfies G(Xp)G(-Xp)T(X)=Xp.
Avitabile, M., Mattarei, S. (2023). The Artin-Hasse series and Laguerre polynomials modulo a prime. AEQUATIONES MATHEMATICAE [10.1007/s00010-023-00996-5].
The Artin-Hasse series and Laguerre polynomials modulo a prime
Avitabile, Marina
;Mattarei, Sandro
2023
Abstract
For an odd prime p, let Ep(X)=∑n=0∞anXn∈Fp[[X]] denote the reduction modulo p of the Artin-Hasse exponential series. It is known that there exists a series G(Xp) ∈ Fp[ [ X] ] , such that Lp-1(-T(X))(X)=Ep(X)·G(Xp) , where T(X)=∑i=1∞Xpi and Lp-1(α)(X) denotes the (generalized) Laguerre polynomial of degree p- 1 . We prove that G(Xp)=∑n=0∞(-1)nanpXnp , and show that it satisfies G(Xp)G(-Xp)T(X)=Xp.
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