TY - JOUR
T1 - A Whipple 7F6 Formula Revisited
AU - Li, Wen Ching Winnie
AU - Long, Ling
AU - Tu, Fang Ting
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media LLC, part of Springer Nature 2022.
PY - 2022/6
Y1 - 2022/6
N2 - A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of ℓ-adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple’s formula to the periods of these modular forms.
AB - A well-known formula of Whipple relates certain hypergeometric values 7F6(1) and 4F3(1). In this paper, we revisit this relation from the viewpoint of the underlying hypergeometric data HD, to which there are also associated hypergeometric character sums and Galois representations. We explain a special structure behind Whipple’s formula when the hypergeometric data HD are primitive and self-dual. If the data are also defined over Q, by the work of Katz, Beukers, Cohen, and Mellit, there are compatible families of ℓ-adic representations of the absolute Galois group of Q attached to HD. For specialized choices of HD, these Galois representations are shown to be decomposable and automorphic. As a consequence, the values of the corresponding hypergeometric character sums can be explicitly expressed in terms of Fourier coefficients of certain modular forms. We further relate the hypergeometric values 7F6(1) in Whipple’s formula to the periods of these modular forms.
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U2 - 10.1007/s44007-021-00015-6
DO - 10.1007/s44007-021-00015-6
M3 - Article
AN - SCOPUS:85195583452
SN - 2730-9657
VL - 1
SP - 480
EP - 530
JO - Matematica
JF - Matematica
IS - 2
ER -