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The Incomplete Gamma Functions Since Tricomi
 In Tricomi's Ideas and Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei
, 1998
"... 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 asy ..."
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Cited by 16 (1 self)
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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.
Harmonic oscillator in characteristic p
 Lett. Math. Phys
, 1998
"... Abstract. We construct an irreducible representation of the canonical commutation relations by operators on a certain Banach space over a local field of characteristic p. The Carlitz polynomials forming the basis of the space are shown to be the counterparts of the Hermite functions for this situati ..."
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Cited by 9 (8 self)
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Abstract. We construct an irreducible representation of the canonical commutation relations by operators on a certain Banach space over a local field of characteristic p. The Carlitz polynomials forming the basis of the space are shown to be the counterparts of the Hermite functions for this situation. The analogues of coherent states are related to the Carlitz exponential. 1.
Fqlinear calculus over function fields
 J. Number Theory
, 1999
"... We define analogues of higher derivatives for Fqlinear functions over the field of formal Laurent series with coefficients in Fq. This results in a formula for Taylor coefficients of a Fqlinear holomorphic function, a definition of classes of Fqlinear smooth functions which are characterized in t ..."
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Cited by 8 (8 self)
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We define analogues of higher derivatives for Fqlinear functions over the field of formal Laurent series with coefficients in Fq. This results in a formula for Taylor coefficients of a Fqlinear holomorphic function, a definition of classes of Fqlinear smooth functions which are characterized in terms of coefficients of their FourierCarlitz expansions. A Volkenborntype integration theory for Fqlinear functions is developed; in particular, an integral representation of the Carlitz logarithm is obtained. Key words: Fqlinear function; Carlitz basis; Carlitz logarithm; Volkenborn integral; difference operator; BargmannFock representation.
Artin automorphisms, Cyclotomic function fields, and Folded listdecodable codes”, http://arxiv.org/abs/0811.4139 17 Venkatesan Guruswami, Anindya C. Patthak, “Correlated AlgebraicGeometric Codes: Improved List Decoding over Bounded Alphabets
 Mathematics of Computation
"... Abstract. Algebraic codes that achieve list decoding capacity were recently constructed by a careful “folding ” of the ReedSolomon code. The “lowdegree ” nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise o ..."
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Cited by 2 (1 self)
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Abstract. Algebraic codes that achieve list decoding capacity were recently constructed by a careful “folding ” of the ReedSolomon code. The “lowdegree ” nature of this folding operation was crucial to the list decoding algorithm. We show how such folding schemes conducive to list decoding arise out of the ArtinFrobenius automorphism at primes in Galois extensions. Using this approach, we construct new folded algebraicgeometric codes for list decoding based on cyclotomic function fields with a cyclic Galois group. Such function fields are obtained by adjoining torsion points of the Carlitz action of an irreducible M ∈ Fq[T]. The ReedSolomon case corresponds to the simplest such extension (corresponding to the case M = T). In the general case, we need to descend to the fixed field of a suitable Galois subgroup in order to ensure the existence of many degree one places that can be used for encoding. Our methods shed new light on algebraic codes and their list decoding, and lead to new codes achieving list decoding capacity. Quantitatively, these codes provide list decoding (and list recovery/soft decoding) guarantees similar to folded ReedSolomon codes but with an alphabet size that is only polylogarithmic in the block length. In comparison, for folded RS codes, the alphabet size is a large polynomial in the block length. This has applications to fully explicit (with no bruteforce search) binary concatenated codes for list decoding up to the Zyablov radius. Contents
Differentiability in local fields of prime characteristic
 Duke Math. J
, 1974
"... function defined on the ring of padic integers is the uniform limit of an interpolation series of binomial form, and he exhibited a necessary and sufficient condition for such a function to be differentiable [2]. In [3] we showed that each continuous linear operator on the ring V of formal power se ..."
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Cited by 2 (0 self)
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function defined on the ring of padic integers is the uniform limit of an interpolation series of binomial form, and he exhibited a necessary and sufficient condition for such a function to be differentiable [2]. In [3] we showed that each continuous linear operator on the ring V of formal power series over a
A comparison of the Carlitz and digit derivative bases in function field arithmetic
 J. Number Theory
, 2000
"... 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 f ..."
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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
A new method and its Application to Generalized qBessel Polynomials
 Department of Mathematics, Uppsala University
, 2001
"... Abstract. We make a brief survey of orthogonal polynomials which are not included in the Askey scheme and we discuss some of their qanalogues. We reintroduce a notation of Heine, which leads to a new method for computations and classifications of qspecial functions. As an example of this method, w ..."
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Abstract. We make a brief survey of orthogonal polynomials which are not included in the Askey scheme and we discuss some of their qanalogues. We reintroduce a notation of Heine, which leads to a new method for computations and classifications of qspecial functions. As an example of this method, we present Bailey’s 1929 transformation formula for a terminating, balanced, verywellpoised 10φ9 qhypergeometric series in the new notation. This series can occur in the Poisson kernel for the AskeyWilson polynomials. We shall conclude by proving a new operator expression for qBessel polynomials and define new associated generalized qBessel polynomials, which will be used to find qanalogues of equations of Chatterjea and Srivastava. Contents
Interpolation series in local fields of prime characteristic
 Duke Math. J
, 1972
"... 1. Introduction. In 1944 Dieudonn [3] proved a padic analogue of the Weierstrass Approximation Theorem by an inductive argument involving the polynomial approximation of certain continuous characteristic functions. In 1958 Mahler [4] proved the sharper result that each continuous padic function de ..."
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1. Introduction. In 1944 Dieudonn [3] proved a padic analogue of the Weierstrass Approximation Theorem by an inductive argument involving the polynomial approximation of certain continuous characteristic functions. In 1958 Mahler [4] proved the sharper result that each continuous padic function defined on the padic integers is the uniform limit of the "interpolation series"
Recent Problems from Uniform Asymptotic Analysis of Integrals In Particular in Connection with Tricomi's $Psi$function
, 1998
"... 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 TricomiCarlitz polynomials in terms of Hermite polynomials. We mention ..."
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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 TricomiCarlitz 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.
KUMMER THEORY OF DIVISION POINTS OVER DRINFELD MODULES OF RANK ONE
, 1999
"... 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 b ..."
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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 nth 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.