Proof of a rational Ramanujan-type series for 1/π . The fastest one in level 3
aa r X i v : . [ m a t h . N T ] N ov PROOF OF A RATIONAL RAMANUJAN-TYPE SERIES FOR /π .THE FASTEST ONE IN LEVEL 3 JES ´US GUILLERA
To Bruce Berndt, in admiration of his inspirational work on Ramanujan’s Notebooks
Abstract.
Using a modular equation of level 3 and degree 23 due to Chan andLiaw, we prove the fastest convergent rational Ramanujan-type series for 1 /π oflevel 3. The method
A prerequisite to understand this paper well is familiarity with the notation andthe method developed in [9]. This method is based on an original idea of Wan [10],whose paper was in turn influenced by some ideas of [8]. Using the notation F s ( α ) = F (cid:18) s , − s (cid:12)(cid:12)(cid:12)(cid:12) α (cid:19) , s ∈ { , , , } , ℓ = 4 sin πs , we proved in [9] the following result: Theorem 1.
Let F s ( α ) = m ( α, β ) F s ( β ) , A ( α, β ) = 0 , be a transformation of modular origin and degree /d , and let β = 1 − α be asolution of A ( α , β ) = 0 , and m = m ( α , β ) . Then, if m = 1 √ d , or m = √ d − ℓ d + √ ℓ d i, we have ∞ X n =0 (cid:0) (cid:1) n (cid:0) s (cid:1) n (cid:0) − s (cid:1) n (1) n ( a + bn ) z n = 1 π , where (1) z = 4 α β , b = 2(1 − α ) r dℓ , a = − α β m ′ α ′ d √ ℓ . or (2) z = 4 α β , b = 2 (1 − α ) r dℓ − , a = − α β m ′ α ′ d √ ℓ , respectively. The ′ means differentiation with respect to the variable that we chooseas independent. The formula ( A, , ∞ X n =0 (cid:0) (cid:1) n (cid:0) (cid:1) n (cid:0) (cid:1) n (1) n (14151 n + 827) ( − n n = 1500 √ π . It is the formula ( A, ,
23) in the notation of [9], we will also refer to it as 3 A
23, andis the fastest convergent rational Ramanujan-type series for 1 /π of level ℓ = 3. It wasdiscovered by Chan, Liaw and Tan [5, eq. 1.19]. As z = 4 α β where β = 1 − α ,we get α = 12 − √ , β = 1 − α = 12 + 53 √ , Then, with a numerical approximation of 20 digits, we have m = F ( α ) F ( β ) ≈ . . i, which we identify as m = √ √ i. Observe that m is of the form m = √ d − ℓ d + √ ℓ d i, | m | = 1 √ d , with d = 23. Hence, for proving (3) with our method, we need a transformation ofdegree 1 /d with d = 23 for the level ℓ = 3, and with such a transformation we canprove it rigorously. This is done in next section using a modular equation in whichthe algebraic relation A ( α, β ) = 0 is written using two auxiliary variables u and v ,in the following way:(4) u k = αβ, v k = (1 − α )(1 − β ) , P ( u, v ) = 0 , where P ( u, v ) is a polynomial in u and v , and k is a positive integer.3. The proof of ( A, , u and v with u and v respectively. It is(5) u = αβ, v = (1 − α )(1 − β ) , P ( u, v ) = 0 , where(6) P ( u, v ) = ( u + v ) − √ u v + uv ) − u v + u v ) − √ u v + u v ) − u v ) − u + v ) − √ u v + uv ) − u v ) + 1 . ROOF OF THE RAMANUJAN-TYPE SERIES 3 A
23 3
If we let β = 1 − α , then we see that u = v . If we choose(7) v = √ − i ! u, and replace it in P ( u, v ) = 0, we have the equation(8) 250(1 + √ i ) u −
952 (1 − √ i ) u + 1 = 0 , which factors as1 + √ i
16 (10 u + 1 + √ i )(20 u − √ − i )(20 u + √ i ) = 0 . One solution is u = √
15 + √
520 + √ − √ i, v = √
15 + √ − √ − √ i. Then, from α (1 − α ) = u = − − , β = 1 − α , we get α = 12 − √ , β = 12 + 531000 √ . Differentiating P ( u, v ) = 0 with respect to u at u = u , we find v ′ = − √ i . Then, differentiating P ( u, v ) = 0 twice with respect to u at u = u , we get v ′′ = 8674041040500000 √ − i ) + 3034180783431000 √ i )17258921684500483 . From (5), we see that(9) u = αβ, u − v + 1 = α + β. Differentiating (9) with respect to u at u = u , we obtain α ′ and β ′ . Then, differen-tiating (9) twice with respect to u at u = u we obtain α ′′ and β ′′ . As the multiplieris given by(10) m = 1 d β (1 − β ) α (1 − α ) α ′ β ′ , see [9], replacing the already known values at u = u , we get(11) m = s α ′ β ′ = √ √ i = √ d − ℓ d + √ ℓ d i. Taking logarithms in (10), differentiating with respect to u , and dividing by α ′ , weget m ′ α ′ = m α ′ (cid:18) β ′ β − β ′ − β − α ′ α + α ′ − α + α ′′ α ′ − β ′′ β ′ (cid:19) , JES ´US GUILLERA and from it, we obtain m ′ α ′ = 82700069 . Finally, using the formulas(12) z = 4 α β , b = 2 (1 − α ) r dℓ − , a = − α β m ′ α ′ d √ ℓ , we obtain z = − , b = 47171500 √ , a = 8274500 √ , and we are done.4. Ramanujan-type series for /π and modular equations In a completely similar way we can prove other Ramanujan-type series for 1 /π using modular equations [1, 2, 3, 4], written in the form (4). For example, for proving( A, , u = αβ, v = (1 − α )(1 − β ) , P ( u, v ) = 0 , where P ( u, v ) = ( u + v ) + 8 √ u v + uv ) + 20 u v − , see [3, 2, Theorem 10.3]. For proving ( A, , u = αβ, v = (1 − α )(1 − β ) , P ( u, v ) = 0 , where P ( u, v ) = ( u + v ) + 3 √ u v + uv ) + 6 u v − . see [3, 2, Theorem 7.8], and we can get the proofs of the formulas ( A, ,
5) and( P, , u = αβ, v = (1 − α )(1 − β ) , u + v + 3 uv − . see [3, 2, Theorem 7.6]. Note that all the above examples correspond to rationalseries. Of course one can use the same method to prove irrational instances.5. Conclusion
As we pointed out in [9], the main aspect of our method is that the modularequations used to prove the Ramanujan-type alternating series for 1 /π have a muchlower degree than those in other methods. It would be interesting to analyze in depththis phenomenon. The reader is invited to compare the values of d in the tables in [9]with the respective values of N given in the tables in [7]. We hope that the methoddeveloped here will attract others to find, for example, new modular equations oflevels ℓ ≥ /π [6, 7]. Of course a generalization of the method would be needed forthat. ROOF OF THE RAMANUJAN-TYPE SERIES 3 A
23 5
References [1]
B.C. Berndt , Ramanujan’s Notebooks, Part III (Springer-Verlag, New York, 1991).[2]
B.C. Berndt , Ramanujan’s Notebooks, Part V (Springer-Verlag, New York, 1998).[3]
B.C. Berndt, S. Bhargava and F.G. Garvan
Ramanujan’s theories of elliptic functionsto alternative bases , Trans. Amer. Soc. , 4163–4244 (1995).[4]
H.H. Chan and W.-C. Liaw , On Russell-Type Modular Equations , Canad. J. Math. ,31–36 (2000).[5] H.H. Chan, W.-C. Liaw and V. Tan , Ramanujan’s class invariant λ n and a new class ofseries for /π , J. London Math Soc. (2) (2001), 93–106.[6] H.H. Chan, S.H. Chan and Zhiguo Liu , Domb’s numbers and Ramanujan–Sato type seriesfor /π , Adv. Math. , 396–410, (2004).[7] H.H. Chan and S. Cooper , Rational analogues of Ramanujan’s series for /π , Math. Proc.Camb. Phil. Soc., 1–23, (2012).[8] H.H. Chan, J. Wan and W. Zudilin , Legendre polynomials and Ramanujan-type series for /π , Israel J. Math. , 183–207, (2013).[9] J. Guillera , A method for proving Ramanujan’s series for /π , Ramanujan J., (to appear).[10] J. Wan , Series for /π Using Legendre’s Relation , Integr. Transforms Spec. Funct. (2014),1-14. University of Zaragoza, Department of mathematics, 50009 Zaragoza (Spain)
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