On the fractal structure of the rescaled evolution set of Carlitz sequences of polynomials

AbstractSelf-similarity properties of the coefficient patterns of the so-called m-Carlitz sequences of polynomials are considered. These properties are coded in an associated fractal set – the rescaled evolution set. We extend previous results on linear cellular automata with states in a finite field. Applications are given for the sequence of Legendre polynomials and sequences associated with the zero Bessel function

The Schur Factorization Property as it Applies to Subsets of the General Laguerre Polynomials

Senior Project submitted to The Division of Science, Mathematics and Computing of Bard College

Schur congruences, Carlitz sequences of polynomials and automaticity

We first generalize the Schur congruence for Legendre polynomials to sequences of polynomials that we call "d-Carlitz". This notion is more general than a similar notion introduced by Carlitz. Then, we study automaticity properties of double sequences generated by these sequences of polynomials, thus generalizing previous results on double sequences produced by one-dimensional linear cellular automata. Keywords: Schur congruence, Legendre and classical polynomials, automatic sequences. 1 Introduction The classical Schur congruence for Legendre polynomials modulo an odd prime p (see [38]) reads: for all n 0 and all v 2 [0; p \Gamma 1], P pn+v j P n (X p )P v (X) mod p: This property is similar to the well-known theorem of Lucas for the binomial coefficients modulo a prime number. We recall Lucas property: for p prime, u and v in [0; p \Gamma 1], and any integers m; n 0, ` pm + u pn + v ' j ` m n '` u v ' mod p: This property can also be written: (1 +X) pn+v j (1 + ..