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Bulletin of the Australian Mathematical Society, ISSN 0004-9727, 08/2018, Volume 98, Issue 1, pp. 27 - 37
We establish some supercongruences for the truncated F-2(1) and F-3(2) hypergeometric series involving the p-adic gamma functions. Some of these results extend... 
supercongruences | truncated hypergeometric series | p-adic gamma functions | LEGENDRE POLYNOMIALS | MATHEMATICS | PRODUCTS | BINOMIAL COEFFICIENTS | HYPERGEOMETRIC-SERIES
Journal Article
Journal of Number Theory, ISSN 0022-314X, 05/2016, Volume 162, pp. 463 - 482
We study a special case of Kashio's p-adic log⁡Γ-function, that we call LogΓp, which combines these of Morita and Diamond. It agrees with each of these on... 
Volkenborn integral | p-adic [formula omitted]-function | P-adic log Γ-function | MATHEMATICS | p-adic log Gamma-function | Mathematics - Number Theory
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 06/2019, Volume 15, Issue 5, pp. 991 - 1007
We study a relation between two refinements of the rank one abelian Gross–Stark conjecture. For a suitable abelian extension H / F of number fields, a... 
The Gross-Stark conjecture | p -adic measures | multiple gamma functions
Journal Article
Journal of Number Theory, ISSN 0022-314X, 08/2012, Volume 132, Issue 8, pp. 1770 - 1779
In this note we study the relationship between the power series expansion of the Dwork exponential and the Mahler expansion of the p-adic Gamma function. We... 
Dwork exponential | p-Adic Gamma function | p-Adic L-functions | Mahler series | P-Adic l-functions | P-Adic gamma function | MATHEMATICS | Universities and colleges
Journal Article
Journal of Number Theory, ISSN 0022-314X, 07/2014, Volume 140, pp. 181 - 195
In [12], McCarthy defined a function Gnn[⋯] using the Teichmüller character of finite fields and quotients of the p-adic gamma function, and expressed the... 
Characters of finite fields | Teichmüller character | p-Adic gamma function | Elliptic curves | Trace of Frobenius | Gauss sums | P-Adic gamma function | Teichmuller character | MATHEMATICS | GAUSSIAN HYPERGEOMETRIC-SERIES
Journal Article
Pacific Journal of Mathematics, ISSN 0030-8730, 2013, Volume 261, Issue 1, pp. 219 - 236
We define a function in terms of quotients of the p-adic gamma function which generalizes earlier work of the author on extending hypergeometric functions over... 
Hypergeometric functions | Elliptic curves | Modular forms | P-adic gamma function | Trace of Frobenius | trace of Frobenius | MATHEMATICS | hypergeometric functions | GAUSSIAN HYPERGEOMETRIC-SERIES | modular forms | FINITE-FIELDS | p-adic gamma function | elliptic curves
Journal Article
Journal of Number Theory, ISSN 0022-314X, 2011, Volume 131, Issue 7, pp. 1240 - 1257
We define p-adic multiple zeta and log gamma functions using multiple Volkenborn integrals, and develop some of their properties. Although our functions are... 
Volkenborn integral | Barnes multiple zeta functions | p-Adic analysis | Multiple gamma functions | Diamond function | P-Adic analysis | MATHEMATICS | VALUES | EULER | Heterocyclic compounds | Universities and colleges
Journal Article
Finite Fields and their Applications, ISSN 1071-5797, 2015, Volume 33, pp. 230 - 231
We correct the statements of Lemma 3.2 and Lemma 3.3 in [1]. 
Characters of finite fields | Teichmüller character | Gauss sums | p-adic Gamma function
Journal Article
Journal of Mathematical Analysis and Applications, ISSN 0022-247X, 03/2019, Volume 471, Issue 1-2, pp. 613 - 622
In 1997, Van Hamme conjectured 13 Ramanujan-type supercongruences labeled (A.2)–(M.2). Using some combinatorial identities discovered by Sigma, we extend (A.2)... 
p-adic Gamma functions | Hypergeometric series | Supercongruences | MATHEMATICS | MATHEMATICS, APPLIED
Journal Article
Bulletin of the Korean Mathematical Society, ISSN 1015-8634, 2013, Volume 50, Issue 2, pp. 583 - 588
In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship between weighted p-adic q-log... 
Modified q-Genocchi numbers with weight alpha and beta | Modified q-Euler numbers with weight alpha and beta | p-adic log gamma functions | POLYNOMIALS | INTEGRALS | MATHEMATICS | modified q-Euler numbers with weight alpha and beta | HIGHER-ORDER | modified q-Genocchi numbers with weight alpha and beta
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 02/2019, Volume 15, Issue 1, pp. 67 - 84
Journal Article
Advances in Mathematics, ISSN 0001-8708, 02/2016, Volume 290, pp. 773 - 808
For the purposes of this paper supercongruences are congruences between terminating hypergeometric series and quotients of p-adic Gamma functions that are... 
p-adic Gamma functions | Hypergeometric series | Supercongruences | MATHEMATICS
Journal Article
Inventiones mathematicae, ISSN 0020-9910, 9/2019, Volume 217, Issue 3, pp. 985 - 1068
We present an integral representation for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other... 
22E50 | Mathematics, general | Mathematics | Primary 11F70 | Secondary 11F55 | 22E55 | 11F66 | GAMMA-FACTORS | CLASSICAL-GROUPS | THEOREM | RANKIN-SELBERG INTEGRALS | BESSEL MODELS | P-ADIC GROUPS | RESIDUAL SPECTRUM | MATHEMATICS | INDUCED REPRESENTATIONS | UNRAMIFIED PRINCIPAL SERIES | FUNCTORIALITY | Mathematical analysis | Tensors | Representations
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 12/2016, Volume 12, Issue 8, pp. 2125 - 2145
The theorems of Gauss and Jacobi that give modulo p evaluations of certain central binomial coefficients have been extended, since the 1980s, to more classes... 
p -adic gamma function | congruences | Catalan numbers | Binomial coefficients | factorials | Jacobi sums | MATHEMATICS | p-adic gamma function | Heterocyclic compounds
Journal Article
Annales mathématiques du Québec, ISSN 2195-4755, 10/2018, Volume 42, Issue 2, pp. 133 - 157
We find summation identities and transformations for the McCarthy’s p-adic hypergeometric series by evaluating certain Gauss sums which appear while counting... 
11S80 | Teichmüller character | Secondary 33C99 | 33E50 | Gaussian hypergeometric series | Mathematics | p -adic Gamma function | Primary 11G20 | Jacobi sums | Algebra | Gauss sums | Analysis | Character of finite fields | p -adic hypergeometric series | Mathematics, general | Algebraic curves | Number Theory | 11T24 | p-adic Gamma function | p-adic hypergeometric series
Journal Article
Journal of the European Mathematical Society, ISSN 1435-9855, 2018, Volume 20, Issue 11, pp. 2813 - 2858
Let H be a connected reductive group defined over a non-archimedean local field F of characteristic p > 0. Using Poincare series, we globalize supercuspidal... 
Local Langlands correspondence | Globalization | Supercuspidal representations | Function fields | GAMMA-FACTORS | local Langlands correspondence | MATHEMATICS, APPLIED | function fields | CLASSICAL-GROUPS | TAMAGAWA NUMBERS | PARAMETERS | CONJECTURE | P-ADIC GROUPS | ALGEBRAIC-GROUPS | MATHEMATICS | DISCRETE-SERIES | REDUCTIVE GROUPS | supercuspidal representations
Journal Article
Bulletin of the Australian Mathematical Society, ISSN 0004-9727, 10/2016, Volume 94, Issue 2, pp. 208 - 216
Journal Article
International Journal of Number Theory, ISSN 1793-0421, 03/2015, Volume 11, Issue 2, pp. 645 - 660
In [The trace of Frobenius of elliptic curves and the p-adic gamma function, Pacific J. Math. 261(1) (2013) 219–236], McCarthy defined a function nGn[⋯] using... 
trace of Frobenius | Character of finite fields | Teichmüller character | elliptic curves | Gaussian hypergeometric series | p-adic Gamma function | Teichmuller character | TRACE | MATHEMATICS | ELLIPTIC-CURVES | GAMMA-FUNCTION | FROBENIUS
Journal Article
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