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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 27 (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 10 (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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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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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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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
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"
Galois groups associated to generic Drinfeld modules and a conjecture of Abhyankar, preprint (2013); arXiv:1303.2334 [math.NT
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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.
On root numbers connected with special values of Lfunctions over Fq
 T ), J. Number Theory
, 1997
"... In this paper we give a few explicit formulae for the dual coefficients and some of the root numbers appearing in the expression of Lfunctions over Fq(T) as s=1. From these formulae we show that the set of root numbers includes all the basic Gauss sums defined by Thakur as a proper subset. We also ..."
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In this paper we give a few explicit formulae for the dual coefficients and some of the root numbers appearing in the expression of Lfunctions over Fq(T) as s=1. From these formulae we show that the set of root numbers includes all the basic Gauss sums defined by Thakur as a proper subset. We also prove some properties of the root numbers. Finally, we provide an example which strongly suggests that there is a close relation between root numbers and the general Gauss sums. 1997 Academic Press 1.