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12
Applications of nonArchimedean integration to the Lseries of τsheaves
 J. Number
"... and Wan, Böckle and Pink [BP1] develop a cohomology theory for F. In [Boc1] Böckle uses this theory to establish the analytic continuation of the Lseries associated to F (which is a characteristic p valued “Dirichlet series”) and the logarithmic growth of the degrees of its special polynomials. In ..."
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and Wan, Böckle and Pink [BP1] develop a cohomology theory for F. In [Boc1] Böckle uses this theory to establish the analytic continuation of the Lseries associated to F (which is a characteristic p valued “Dirichlet series”) and the logarithmic growth of the degrees of its special polynomials. In this paper we shall show that this logarithmic growth is all that is needed to analytically continue the original Lseries as well as all associated partial Lseries. Moreover, we show that the degrees of the special polynomials attached to the partial Lseries also grow logarithmically. Our tools are Böckle’s original results, nonArchimedean integration, and the very strong estimates of Y. Amice [Am1]. Along the way, we define certain natural modules associated with nonArchimedean measures (in the characteristic 0 case as well as in characteristic p). 1.
Umbral calculus in positive characteristic
 Adv. Appl. Math
"... Partially supported by CRDF under Grant UM12567OD03 An umbral calculus over local fields of positive characteristic is developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. Orthonormal bases in the space of continuous Fqlinear functions are constructed. Key ..."
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Partially supported by CRDF under Grant UM12567OD03 An umbral calculus over local fields of positive characteristic is developed on the basis of a relation of binomial type satisfied by the Carlitz polynomials. Orthonormal bases in the space of continuous Fqlinear functions are constructed. Key words: Fqlinear function; delta operator; basic sequence; orthonormal basis 2 1
Polylogarithms and a zeta function for finite places of a function field, Contemporary Math
"... We introduce and study new versions of polylogarithms and a zeta function on a completion of Fq(x) at a finite place. The construction is based on the use of the Carlitz differential equations for Fqlinear functions. Key words: Fqlinear function; polylogarithms; zeta function 2 1 ..."
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We introduce and study new versions of polylogarithms and a zeta function on a completion of Fq(x) at a finite place. The construction is based on the use of the Carlitz differential equations for Fqlinear functions. Key words: Fqlinear function; polylogarithms; zeta function 2 1
Hypergeometric Functions and Carlitz Differential Equations over Function Fields
, 2005
"... The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur’s hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations wi ..."
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The paper is a survey of recent results in analysis of additive functions over function fields motivated by applications to various classes of special functions including Thakur’s hypergeometric function. We consider basic notions and results of calculus, analytic theory of differential equations with Carlitz derivatives (including a counterpart of regular singularity), umbral calculus, holonomic modules over the WeylCarlitz ring.
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
4 PADIC INTERPOLATION AND MULTIPLICATIVE ORIENTATIONS OF KO AND TMF (WITH AN APPENDIX BY NIKO NAUMANN)
"... ar ..."
DworkCarlitz Exponential and Overconvergence for Additive Functions in Positive Characteristic
, 2006
"... We study overconvergence phenomena for Fqlinear functions on a function field over a finite field Fq. In particular, an analog of the Dwork exponential is introduced. ..."
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We study overconvergence phenomena for Fqlinear functions on a function field over a finite field Fq. In particular, an analog of the Dwork exponential is introduced.
CARLITZ EXTENSIONS
"... 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 aft ..."
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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