Eisenstein series and Ramanujan-type series for 1/π

Baruah, Nayandeep Deka; Berndt, Bruce C.

doi:10.1007/s11139-008-9155-8

Cited by 38 publications

(4 citation statements)

References 30 publications

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“…In 2005 Y. Yang used modular forms of level 10 to discover the following curious identity relating Franel numbers of order 4 to Ramanujan-type series This has not been published by Yang, but more identities of this kind were deduced by S. Cooper [4] in 2012 via modular forms. For the classical Ramanujan-type series for 1/π, one may consult [1,2,10] and the nice survey by Cooper [5,Chapter 14].…”

Section: Introductionmentioning

confidence: 99%

Some new series for $1/\pi $ motivated by congruences

Sun

2023

Colloq. Math.

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Section: Introductionmentioning

confidence: 99%

Some new series for $1/\pi $ motivated by congruences

Sun

2023

Colloq. Math.

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“…where b, c, m are integers with bm = 0, µ is a positive and square free integer, λ is a nonzero number, and a(n) is a product of binomial coefficients in { 2n Extensive work has been done around applications, q-ananlogue forms and p-adic analogue forms of Ramaujan-type series. One can consult [2,3,7,[9][10][11][19][20][21][22] and references therein for more related results.…”

Section: Introductionmentioning

confidence: 99%

Dynamics Characteristics of Groundwater Exploitation in Tianjin, P.R. China

Wang

Wei

et al. 2014

AMR

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We characterize observable sets for 1-dim Schrödinger equations in R: i∂ t u = (−∂ 2x + x 2m )u (with m ∈ N := {0, 1, . . . }). More precisely, we obtain what follows: First, when m = 0, E ⊂ R is an observable set at some time if and only if it is thick, namely, there is γ > 0 and L > 0 so thatSecond, when m = 1 (m ≥ 2 resp.), E is an observable set at some time (at any time resp. ) if and only if it is weakly thick, namelyFrom these, we see how potentials x 2m affect the observability (including the geometric structures of observable sets and the minimal observable time). Besides, we obtain several supplemental theorems for the above results, in particular, we find that a half line is an observable set at time T > 0 for the above equation with m = 1 if and only if T > π 2 .2010 Mathematics Subject Classification. 93B07, 35J10.

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“…The classical rational Ramanujan-type series for π −1 (cf. [1,2,8,27] and a nice introduction by S. Cooper [10,Chapter 14]) have the form In 1997 Van Hamme [47] conjectured that such a series ( * ) has a p-adic analogue of the form…”

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confidence: 99%

“…. , 10) are 1,2,10,68,586,5252,49204,475400,4723786,47937812,494786260 respectively. We may extend the numbers S n (n ∈ N) further.…”

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confidence: 99%