Arithmetic hypergeometric series

Abstract

The main goal of our survey is to give common characteristics of auxiliary hypergeometric functions (and their generalisations), functions which occur in number-theoretical problems. Originally designed as a tool for solving these problems, the hypergeometric series have become a connecting link between different areas of number theory and mathematics in general. Bibliography: 183 titles.

Linear Mahler measures and double L-values of modular forms

“…Conjecture (F. Rodriguez-Villegas, [BLVD], see also [Zud1]): This modular form arises in [PTV] in relation to the variety 1 + x 1 + x 2 + x 3 + x 4 = 0 1 + 1 x1 + 1 x2 + 1 x3 + 1 x4 = 0 which can be compactified to a K3 surface of Picard rank 20. Namely, C. Peters, J.…”

Two hypergeometric tales and a new irrationality measure of $$\zeta (2)$$ ζ ( 2 )

“…Though it will not be as important as it was in our arithmetic consideration of Section 3, we introduce the ordered versionsâ * 1 ≤â * 2 ≤â * 3 of the parametersâ 1 ,â 2 ,â 3 andb * 2 ≤b * 3 ofb 2 ,b 3 . Then this ordering and conditions (20) imply that the rational functionR(t) has poles at t = −k forâ * 2 ≤ k ≤b * 3 − 1, double poles at t = −k forâ * 3 ≤ k ≤b * 2 − 1, and zeroes at t = −ℓ/2 forb 0 ≤ ℓ ≤â * 0 − 1 whereâ * 0 = min{â 0 , 2â * 2 }. The partial-fraction decomposition of the regular rational functionR(t) assumes the formR…”

“…dt. (19) The satellite identity, in which (π/ sin πt) 2 and π/ sin 2πt are replaced with π 3 cos πt/ (sin πt) 3 and (π/ sin πt) 2 , respectively, is expected to hold as well; the other integrals represent rational approximations to ζ(3) [4,20]. These identities can be possibly shown in full generality using contiguous relations for the integrals on both sides; it seems to be a tough argument though.…”

“…. , 3, satisfy 1 2β 0 ,β 1 < 1 2α 0 ,α 1 ,α 2 ,α 3 <β 2 ,β 3 ,α 0 +α 1 +α 2 +α 3 =β 0 +β 1 +β 2 +β 3 to ensure that all hypotheses (20) are satisfied. Though we can give the analogue of Lemma 6, our principal interest in the construction of this section is purely arithmetic.…”

Hall–Littlewood polynomials and characters of affine Lie algebras

“…Andrews' multiple series transformation [2] is one of the most complicated results in all of the theory of basic hypergeometric series. It is also one of the most useful; it implies many important partition and Rogers-Ramanujan-type identities [2] and has recently played a major role in answering deep arithmetic questions related to the Riemann zeta function, see e.g., [26,33,34,77].…”