The aim of this note is to provide a natural extension and generalisation of the well-known Gelfond constant eπ using a hypergeometric function approach. An extension is also found for the square root of this constant. Several known mathematical constants are also deduced in hypergeometric form from our newly introduced constant

Milovanović, Gradimir V.

Paris, Richard B.

Rathie, Arjun K.

English

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MATEMATIČKI VESNIKMATEMATIQKI VESNIK74, 1 (2022), 71–77March 2022research paperoriginalni nauqni radHYPERGEOMETRIC REPRESENTATIONS OF GELFOND’SCONSTANT AND ITS GENERALISATIONSArjun K. Rathie, Gradimir V. Milovanović and Richard B. ParisAbstract. The aim of this note is to provide a natural extension and generalisation ofthe well-known Gelfond constant eπ using a hypergeometric function approach. An extensionis also found for the square root of this constant. Several known mathematical constants arealso deduced in hypergeometric form from our newly introduced constant.1. Introduction and preliminariesIn mathematics, Gelfond’s constant, which is named after Aleksandr Osipovich Gel-fond (1906–1968), is given by eπ. Like both e and π, this constant is a transcendentalnumber. The decimal expansion of Gelfond’s constant iseπ = 23.1406926 32779 . . .and its continued fraction representation is given in [5, A039661].This number has a connection to the Ramanujan constant eπ√163 = (eπ)√163. Itis worth noting that this last number is almost an integer:eπ√163 ≃ 6403203 + 744.A geometrical occurrence of Gelfond’s constant arises in the sum of even-dimensionunit spheres with volume V2n = πn/n!. Then∞∑n=0V2n = eπ.There are several ways of expressing Gelfond’s constant, some of which are enu-merated below. We have eπ = (ii)−2, where i =√−1;eπ = 0F1(−−12∣∣∣∣ π24)+ π 0F1(−−32∣∣∣∣ π24),2020 Mathematics Subject Classification: 11Y60, 33B10, 33C05, 33C20.Keywords and phrases: Gelfond’s constant; hypergeometric function; Gauss summation the-orem.7172 Hypergeometric representations of Gelfond’s constantwhere 0F1 is a generalised hypergeometric function that can be expressed in termsof the modified Bessel function of the first kind of order ∓1/2 (i.e., as πI−1/2(π)/√2and I1/2(π)/√2, resp.) and finallyeπ = 2F1(i, −i12∣∣∣∣ 1)+ 2 2F1( 12 + i, 12 − i32∣∣∣∣ 1), (1)where 2F1 is the well-known Gauss hypergeometric function [3, p. 384]. The result (1)can be easily verified by making use of the classical Gauss summation theorem2F1(a, bc∣∣∣∣ 1) = ∞∑n=0(a)n(b)n(c)n1n!=Γ(c)Γ(c− a− b)Γ(c− a)Γ(c− b), (2)where (a)n = Γ(a + n)/Γ(a) is Pochhammer’s symbol and Re(c − a − b) > 0, toevaluate the 2F1 series on the right-hand side, combined with the identitiescosπz =πΓ( 12 − z)Γ(12 + z), sinπz =πΓ(z)Γ(1− z)and Euler’s formula eiz = cos z + i sin z.The extension of the summation theorem (2) to the 3F2 hypergeometric series isavailable in the literature [4], which we write in the following manner:3F2(a, b, d+ 1c+ 1, d∣∣∣∣ 1) = ∞∑n=0(a)n(b)b(d+ 1)n(c+ 1)n(d)n1n!=Γ(c+ 1)Γ(c− a− b)Γ(c− a+ 1)Γ(c− b+ 1){c− a− b+ abd}(3)provided d ̸= 0,−1,−2, . . . and Re(c− a− b) > 0.The aim of this note is to provide an extension of Gelfond’s constant (1), andalso its square root, with the help of the result (3). A few interesting results closelyrelated to Gelfond’s constant and its square root are also given.2. Extension of Gelfond’s constantThe natural extension of Gelfond’s constant to be established here is given in thefollowing theorem.Theorem 2.1. For d1, d2 ̸= 0,−1,−2, . . . , the following result holds true:eπ(15d1+1532d2+2380)+ e−π(15d1− 1532d2− 780)= 3F2(i, −i, d1 + 132 , d1∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, d2 + 152 , d2∣∣∣∣ 1). (4)Proof. The derivation of (4) follows from application of the summation formula (3).We have3F2(i, −i, d1 + 132 , d1∣∣∣∣ 1) = (eπ + e−π)( 110 + 15d1)A. K. Rathie, G. V. Milovanović, R. B. Paris 73and3F2( 12 + i,12 − i, d2 + 152 , d2∣∣∣∣ 1) = (eπ − e−π)( 332 + 1564d2).Insertion of these summations into the right-hand side of (4) then yields the resultasserted by the theorem. □Corollary 2.2. In (4), if we take d1 = 2/(5n − 1) and d2 = 15/(2(8n − 3)) forpositive integer n, then we obtain after a little calculation the following result:neπ = 3F2(i, −i, 5n+15n−132 ,25n−1∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 16n+92(8n−3)52 ,152(8n−3)∣∣∣∣ 1). (5)For n = 1, 2, 3 we find respectively the following results related to (1):eπ = 3F2(i, −i, 3232 ,12∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 5252 ,32∣∣∣∣ 1),2eπ = 3F2(i, −i, 11932 ,29∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 412652 ,1526∣∣∣∣ 1),3eπ = 3F2(i, −i, 8732 ,17∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 191452 ,514∣∣∣∣ 1).We note that the expression for eπ reduces to (1) since both 3F2 series contract toyield 2F1 series.Corollary 2.3. In (4), if we take d1 = 2/(5n − 1) and d2 = −15/(2(8n + 3)) forpositive integer n, then we obtain after a little calculation the following result:ne−π = 3F2(i, −i, 5n+15n−132 ,25n−1∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 16n−92(8n+3)32 , −152(8n+3)∣∣∣∣ 1).In particular, for n = 1, 2, 3 we find respectively the following results:e−π =3F2(i, −i, 3232 ,12∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 72232 , −1522∣∣∣∣ 1),2e−π =3F2(i, −i, 11932 ,29∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 233832 , −1538∣∣∣∣ 1),3e−π =3F2(i, −i, 8732 ,17∣∣∣∣ 1)+ 2 3F2( 12 + i, 12 − i, 131832 , −518∣∣∣∣ 1),where the first 3F2 series in the expression for e−π contracts to yield a simpler 2F1series.Similarly, if we take d1 = 2/(10n− 1) and d2 = −5/2 in (4), where n is a positiveinteger, and note that the second 3F2 series vanishes with this choice of d2 on accountof (3), we obtain after a little calculation the following:Corollary 2.4. We have the result for positive integer nn(eπ + e−π) = 3F2(i, −i, 10n+110n−132 ,210n−1∣∣∣∣ 1).74 Hypergeometric representations of Gelfond’s constantIn particular, for n = 1, 2 we find the following results:eπ + e−π = 3F2(i, −i, 11932 ,29∣∣∣∣ 1), 2(eπ + e−π) = 3F2( i, −i, 211932 ,219∣∣∣∣ 1).Similarly other results can be obtained.3. The square root of Gelfond’s constant: eπ/2Expressions for the square root of Gelfond’s constant are eπ/2 = i−i,eπ/2 = 2F1(i, −i12∣∣∣∣ 12)+√2 2F1(12 + i,12 − i32∣∣∣∣ 12)(6)together with the inverse expressione−π/2 = 2F1(i, −i12∣∣∣∣ 12)−√2 2F1(12 + i,12 − i32∣∣∣∣ 12). (7)It is not out of place to mention here that the results in (6) and (7) can be obtainedby evaluating the first hypergeometric function by the second Gauss theorem and thesecond hypergeometric function by Bailey’s theorem (cf. [2])2F1(a, b12 (a+ b+ 1)∣∣∣∣ 12)=Γ(12)Γ(12a+12b+12)Γ(12a+12)Γ(12b+12) ,2F1(a, 1− ac∣∣∣∣ 12)=Γ(12c)Γ(12c+12)Γ(12c+12a)Γ(12c−12a+12) .We now derive the analogue of Theorem 2.1 by making use of the extension of thesecond Gauss and Bailey’s theorems applied to 3F2 series. These are given by [1]:3F2(a, b, d+112 (a+b+3), d∣∣∣∣ 12)=Γ(12)Γ(12a+12b+32)Γ(12a−12b−12)Γ(12a−12b+32)×{ 12 (a+b−1)−ab/dΓ(12a+12)Γ(12b+12)+(a+b+1)/d−2Γ(12a)Γ(12b) }, (8)3F2(a, 1−a, d+1c+1, d∣∣∣∣ 12)= 2−cΓ(12)Γ(c+1)×{2/dΓ(12c+12a)Γ(12c−12a+12)+ 1−(c/d)Γ(12c+12a+12)Γ(12c−12a+1)}, (9)provided d ̸= 0,−1,−2, . . . . Then we have the following theorem.Theorem 3.1. For d1, d2 ̸= 0,−1,−2, . . . , the following result holds true:eπ/2(110d1+316d2+2740)+ e−π/2(310d1− 2116d2+1140)= 3F2(i, −i, d1 + 132 , d1∣∣∣∣ 12)+√2 3F2( 12 + i,12 − i, d2 + 152 , d2∣∣∣∣ 12). (10)A. K. Rathie, G. V. Milovanović, R. B. Paris 75Proof. In the first 3F2 series use (8) and in the second 3F2 series use (9) together withstandard properties of the gamma function. □Corollary 3.2. If in (10) we take d1 = 1/(7n − 5) and d2 = 15/(24n − 14) forpositive integer n then we findneπ/2 = 3F2(i, −i, 7n−47n−532 ,17n−5∣∣∣∣ 12)+√2 3F2( 12 + i,12 − i,24n+124n−1452 ,1524n−14∣∣∣∣ 12).When n = 1 we recover (6) after contraction of the 3F2 series. For n = 2, 3 wefind respectively the following results:2eπ/2 =3F2(i, −i, 10932 ,19∣∣∣∣ 12)+√2 3F2( 12 + i,12 − i,493452 ,1534∣∣∣∣ 12),3eπ/2 =3F2(i, −i, 171632 ,116∣∣∣∣ 12)+√2 3F2( 12 + i,12 − i,735852 ,1558∣∣∣∣ 12).Similarly other results can be obtained.4. Generalisations of Gelfond’s constantThe natural generalisation of Gelfond’s constant established in this section is givenby the following theorem.Theorem 4.1. For µ > 0 and ν an arbitrary complex parameter with λ = π−1 ln µ+ν,the following result holds true:µeπν = eπλ = 2F1(iλ, −iλ12∣∣∣∣ 1)+ 2λ 2F1( 12 + iλ, 12 − iλ32∣∣∣∣ 1) (11)Proof. The proof of (11) follows along similar lines to that outlined for (1). □Remark 4.2. Other ways of expressing eπλ are:eπλ = (ii)−2λ (i =√−1); eπλ = 0F1(−−12∣∣∣∣ π2λ24)+ πλ 0F1(−−32∣∣∣∣ π2λ24),Remark 4.3. The result (11) yields the alternative expression for e±π/2 given bye±π/2 = 2F1( 12 i, −12 i12∣∣∣∣ 1)± 2F1( 12 + 12 i, 12 − 12 i32∣∣∣∣ 1).Here we mention several mathematical constants that are derivable from our newconstant µeπν by suitable choice of the parameters µ and ν. Each constant canconsequently be expressed in terms of Gauss hypergeometric functions by (11). Theseconstants are enumerated below.If we choose µ = 1, ν =√19,√43,√67 and√163 in (11), we obtain hypergeo-metric function representations of the four largest Heegner numbers, viz.eπ√19 = 8.85479 77768 01543 19497 . . .× 105 .= 963 + 744− 0.22;76 Hypergeometric representations of Gelfond’s constanteπ√43 = 8.84736 74399 97774 66034 . . .× 108 .= 9603 + 744− 0.00022;eπ√67 = 1.47197 95274 39999 98662 . . .× 1011 .= 52803 + 744− 0.0000013;eπ√163 = 2.62537 41264 07687 43999 . . .× 1017 .= 6403203 + 744− 7.5× 10−13,the last number being Ramanujan’s constant.Other choices for µ and ν in (11) yield hypergeometric representations of thefollowing constants:µ = 1, ν = 1 : Gelfond eπ = 23.14069 26327 79269 00572 . . . ;µ = π2/8, ν = 0 : Favard34ζ(2) = 1.23370 05501 36169 82735 . . . ;µ = π2/12, ν = 0 : Nielsen-Ramanujan12ζ(2) = 0.82246 70334 24113 21823 . . . ;µ =Γ(13)Γ(56)Γ(16) , ν = 0 : Bloch-LandauL = 0.54325 89653 42976 70695 . . . ;µ = 2√2, ν = 0 : Gelfond-SchneiderGGS = 2.66514 41426 90225 18865 . . . ;µ =π212 log 2, ν = 0 : Khinchin-Lévyβ = 1.18656 91104 15625 45282 . . . ;µ =4√π23/4Γ2(14) , ν = 18: Weierstrassσ(12)= 0.47494 93799 87920 65033 . . . ;µ =πlog 2, ν = 0 : Van der Pauw α = 4.53236 01418 27193 80962 . . . ;µ =6π2log 2 log 10, ν = 0 : Loche LLo = 0.97027 01143 92033 92574 . . . ;µ =14π3/2Γ2(14), ν = 0 : Chebyshev λCh = 0.59017 02995 08048 11302 . . . .Similarly several other constants available in the literature can be easily found from (11).Acknowledgement. The work of G.V. Milovanović was supported in part bythe Serbian Academy of Sciences and Arts (Project Φ-96).References[1] Y.S. Kim, M.A. Rakha, A.K. Rathie, Extensions of certain classical summation theorems forthe series 2F1, 3F2 and 4F3 with applications in Ramanujan’s summations, Int. J. Math. Sci.2010, Article ID 309503, 26 pages.[2] G.V. Milovanović, R.K. Parmar, A.K. Rathie, A study of generalized summation theoremsfor the series 2F1 with an applications to Laplace transforms of convolution type integralsinvolving Kummer’s functions 1F1, Appl. Anal. Discrete Math. 12 (2018), 257–272.A. K. Rathie, G. V. Milovanović, R. B. Paris 77[3] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark (eds.), NIST Handbook of MathematicalFunctions, Cambridge University Press, Cambridge, 2010.[4] A.P. Prudnikov, Yu. A. Brychkov, O.I. Marichev, Integrals and Series: Special Functions,Vol. 3, Gordon and Breach, New York, 1988.[5] N.J.A. Sloane, On-line Encyclopedia of Integer Sequences, 2008 (https://oeis.org).(received 10.03.2021; in revised form 29.04.2021; available online 31.12.2021)Department of Mathematics, Vedant College of Engineering and Technology, RajasthanTechnical University, Bundi, 323021, Rajasthan, IndiaE-mail: arjunkumarrathie@gmail.comSerbian Academy of Sciences and Arts, 11000 Beograd, SerbiaFaculty of Science and Mathematics, University of Nǐs, 18000 Nǐs, SerbiaE-mail: gvm@mi.sanu.ac.rsDivision of Computing and Mathematics, Abertay University, Dundee DD1 1HG, UKE-mail: r.paris@abertay.ac.uk

Hypergeometric representations of Gelfond's constant and its generalisations

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Hypergeometric representations of Gelfond's constant and its generalisations

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