We define analogues of higher derivatives for Fq-linear functions over the field of formal Laurent series with coefficients in Fq. This results in a formula for Taylor coefficients of a Fq-linear holomorphic function, a definition of classes of Fq-linear smooth functions which are characterized in terms of coefficients of their Fourier-Carlitz expansions. A Volkenborn-type integration theory for Fq-linear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained. Key words: Fq-linear function; Carlitz basis; Carlitz logarithm; Volkenborn integral; difference operator; Bargmann-Fock representation.

The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

Abstract. We make a brief survey of orthogonal polynomials which are not included in the Askey scheme and we discuss some of their q-analogues. We reintroduce a notation of Heine, which leads to a new method for computations and classifications of q-special functions. As an example of this method, we present Bailey’s 1929 transformation formula for a terminating, balanced, verywell-poised 10φ9 q-hypergeometric series in the new notation. This series can occur in the Poisson kernel for the Askey-Wilson polynomials. We shall conclude by proving a new operator expression for q-Bessel polynomials and define new associated generalized q-Bessel polynomials, which will be used to find q-analogues of equations of Chatterjea and Srivastava. Contents

The paper discusses asymptotic methods for integrals, in particular uniform approximations. We discuss several examples, where we apply the results to Tricomi's \Psi\Gammafunction, in particular we consider an expansion of Tricomi-Carlitz polynomials in terms of Hermite polynomials. We mention other recent expansions for orthogonal polynomials that do not satisfy a differential equation, and for which methods based on integral representations produce powerful uniform asymptotic expansions. 1991 Mathematics Subject Classification: 41A60, 33B20, 33C10, 33C45, 11B73, 30E15. Keywords and Phrases: uniform asymptotic expansions, Tricomi's \Psi\Gammafunction, Kummer functions, confluent hypergeometric functions, Whittaker functions, Hermite polynomials, Tricomi-Carlitz polynomials. Note: Work carried out under project MAS2.8 Exploratory research. Extended version of a paper presented at the Conference Tricomi's Ideas and Contemporary Applied Mathematics to celebrate the 100th anniversary o...

Abstract: We compare several properties and constructions of the Carlitz polynomials and digit derivatives for continuous functions on Fq[[T]]. In particular, we show a close relation between them as orthonormal bases. Moreover, parallel to Carlitz’s coefficient formula, we give the closed formula for the expansion coefficients in terms of the digit derivatives. 1

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The ring Z has many analogies with the ring Fp[T], where Fp is a field of prime size p. For example, for nonzero m ∈ Z and M ∈ Fp[T], the residue rings Z/(m) and Fp[T]/M are both finite. The unit groups Z × = {±1} and Fp[T] × = F × p are both finite. Every nonzero integer can be made positive after multiplication by a suitable sign ±1, and every

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We propound a Descent Principle by which previously constructed equations over GF(q n )(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups, where q = p u ? 1 is a power of a prime p and n is a positive integer. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degree m with Galois group GL(m; q) where m is any positive integer and then, under suitable conditions, enlarging its Galois group to GL(m; q n ) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. So, alternatively, we may regard this as an Ascent Principle. Elsewhere we proved this when m is square-free with GCD(mnu; 2p) = 1 = GCD(m;n). There the proof was based on CT (= the Classification Theorem of Finite Simple Groups) in its incarnation of CPT (= the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). Here, without ...

Abstract. A Kummer theory of division points over rank one Drinfeld A = Fq[T]- modules defined over global function fields was given. The results are in complete analogy with the classical Kummer theory of division points over the multiplicative algebraic group Gm defined over number fields. Let K be a number field and let ¯ K be a fixed algebraic closure of K. For any positive integer n, let µn be the group of n-th roots of unity in ¯ K. Let G(n) = Gal(K(µn)/K). For K = Q, G(n) ∼ = (Z/nZ) ∗ , and for any number field K, G(l) ∼ = (Z/lZ) ∗ for almost all prime numbers l.