A note on an extension of Gelfond's constant
aa r X i v : . [ m a t h . C A ] J u l A note on an extension of Gelfond’s constant
Arjun K. Rathie a and Richard B. Paris ba Department of Mathematics, Vedant College of Engineering and Technology(Rajasthan Technical University), Bundi, 323021, Rajasthan, IndiaE-Mail: [email protected] b Division of Computing and Mathematics,Abertay University, Dundee DD1 1HG, UKE-Mail: [email protected]
Abstract
The aim of this note is to provide a natural extension of Gelfond’s constant e π using ahypergeometric function approach. An extension is also found for the square root of thisconstant. A few interesting special cases are presented. Mathematics Subject Classification:
Keywords:
Gelfond’s constant, hypergeometric function, Gauss summation theorem.
1. Introduction
In mathematics, Gelfond’s constant, which is named after Aleksandr Gelfond, is given by e π .Like both e and π , this constant is a transcendental number. The decimal expansion of Gelfond’sconstant is e π = 23 . . . . and its continued fraction representation is given in [4, A039661].This number has a connection to the Ramanujan constant e π √ = ( e π ) √ . It is worthnoting that this last number is almost an integer: e π √ ≃ + 744 . A geometrical occurrence of Gelfond’s constant arises in the sum of even-dimension unit sphereswith volume V n = π n /n !. Then ∞ X n =0 V n = e π . There are several ways of expressing Gelfond’s constant, some of which are enumeratedbelow: e π = ( i i ) − ( i = √− A.K. Rathie and R.B. Paris e π = ∞ X k =0 ( − k k ! ! − s , s = ∞ X k =0 ( − k k + 1 ; e π = (cid:18) ∞ Y k =1 k − µ ( k ) /k (cid:19) σ , σ = p (1) , where µ ( k ) is the M¨obius function and Li n ( x ) is the polylogarithm function; e π = F ( −− ; ; π /
4) + π F ( −− ; ; π / , where F ( −− ; a ; z ) is a generalised hypergeometric function that can be expressed in terms ofmodified I -Bessel functions of order ± ; and finally e π = F ( i, − i ; ; 1) + 2 F ( + i, − i ; ; 1) , (1.1)where F ( a, b ; c ; z ) is the well-known Gauss hypergeometric function [2, p. 384]. The result(1.1) can be easily established by making use of the classical Gauss summation theorem F ( a, b ; c ; 1) = Γ( c )Γ( c − a − b )Γ( c − a )Γ( c − b ) (1.2)provided ℜ ( c − a − b ) > F hypergeometric series isavailable in the literature [3], which we shall write in the following manner: F (cid:18) a, b, d + 1 c + 1 , d ; 1 (cid:19) = Γ( c + 1)Γ( c − a − b )Γ( c − a + 1)Γ( c − b + 1) (cid:26) c − a − b + abd (cid:27) (1.3)provided d = 0 , − , − , . . . and ℜ ( c − a − b ) >
0. The aim of this note is to provide a naturalextension of Gelfond’s constant (1.1), and also its square root, with the help of the result (1.3).A few interesting results closely related to Gelfond’s constant and its square root are also given.
2. Extension of Gelfond’s constant
The natural extension of Gelfond’s constant to be established here is given in the followingtheorem.
Theorem 1
For d , d = 0 , − , − , . . . , the following result holds true: e π (cid:18) d + 1532 d + 2380 (cid:19) + e − π (cid:18) d − d − (cid:19) = F (cid:18) i, − i, d + 1 , d ; 1 (cid:19) + 2 F (cid:18) + i, − i, d + 1 , d ; 1 (cid:19) . (2.1) Proof.
The derivation of (2.1) follows from application of the summation formula (1.3). Wehave F (cid:18) i, − i, d + 1 , d ; 1 (cid:19) = ( e π + e − π ) (cid:18)
110 + 15 d (cid:19) elfond’s constant F (cid:18) + i, − i, d + 1 , d ; 1 (cid:19) = ( e π − e − π ) (cid:18)
332 + 1564 d (cid:19) . Insertion of these summations into the right-hand side of (2.1) then yields the result assertedby the theorem ✷
3. Corollaries
In this section, we mention some interesting special cases of our main result in (2.1).
Corollary 1
In (2.1), if we take d = 2 / (5 n − and d = 15 / (2(8 n − for positive integer n , then we obtain after a little calculation the following result: ne π = F (cid:18) i, − i, n +15 n − , n − ; 1 (cid:19) + 2 F (cid:18) + i, − i, n +92(8 n − , n − ; 1 (cid:19) . (3.1)In particular, when n = 1 we recover Gelfond’s constant (1.1). For n = 2 , e π = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, , ; 1 (cid:19) (3.2)and 3 e π = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, , ; 1 (cid:19) . (3.3) Corollary 2
In (2.1), if we take d = 2 / (5 n − and d = − / (2(8 n + 3)) for positiveinteger n , then we obtain after a little calculation the following result: ne − π = F (cid:18) i, − i, n +15 n − , n − ; 1 (cid:19) + 2 F (cid:18) + i, − i, n − n +3)32 , − n +3) ; 1 (cid:19) . (3.4)In particular, for n = 1 , , e − π = F (cid:18) i, − i ; 1 (cid:19) + 2 F (cid:18) + i − i, , − ; 1 (cid:19) , (3.5)2 e − π = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, , − ; 1 (cid:19) (3.6)and 3 e − π = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, , − ; 1 (cid:19) . (3.7) A.K. Rathie and R.B. Paris
Corollary 3
In (2.1), if we take d = 1 / (2(10 n − and d = − / for positive integer n ,then we obtain after a little calculation the following result: n ( e π + e − π ) = F (cid:18) i, − i, n − n − , n − ; 1 (cid:19) + 2 F (cid:18) + i, − i, − , − ; 1 (cid:19) . (3.8)In particular, for n = 1 , , e π + e − π = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, − , − ; 1 (cid:19) , (3.9)2( e π + e − π ) = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, − , − ; 1 (cid:19) (3.10)and 3( e π + e − π ) = F (cid:18) i, − i, , ; 1 (cid:19) + 2 F (cid:18) + i, − i, − , − ; 1 (cid:19) . (3.11)Similarly other results can be obtained.
4. The square root of Gelfond’s constant: e π/ Expressions for the square root of Gelfond’s constant are: e π/ = i − i ; e π/ = ∞ X k =0 ( − k k ! ! − s , s = ∞ X k =0 ( − k k + 1 ; e π/ = F (cid:18) i, − i ; 12 (cid:19) + √ F (cid:18) + i, − i ; 12 (cid:19) (4.1)together with the inverse expression e − π/ = F (cid:18) i, − i ; 12 (cid:19) − √ F (cid:18) + i, − i ; 12 (cid:19) . (4.2)The results in (4.1) and (4.2) can be obtained by evaluating the first hypergeometric functionby the second Gauss theorem and the second hypergeometric function by Bailey’s theorem viz. F (cid:18) a, b ( a + b + 1) ; 12 (cid:19) = Γ( )Γ( a + b + )Γ( a + )Γ( b + ) , F (cid:18) a, − ac ; 12 (cid:19) = Γ( c )Γ( c + )Γ( c + a )Γ( c − a + ) . We now derive the analogue of Theorem 1 by making use of the extension of the secondGauss and Bailey’s theorems applied to F series. These are given by [1]: F (cid:18) a, b, d + 1 ( a + b + 3) , d ; 12 (cid:19) = Γ( )Γ( a + b + )Γ( a − b − )Γ( a − b + ) elfond’s constant × (cid:26) ( a + b − − ab/d Γ( a + )Γ( b + ) + ( a + b + 1) /d − a )Γ( b ) (cid:27) (4.3)and F (cid:18) a, − a, d + 1 c + 1 , d ; 12 (cid:19) = 2 − c Γ( )Γ( c + 1) × (cid:26) /d Γ( c + a )Γ( c − a + ) + 1 − ( c/d )Γ( c + a + )Γ( c − a + 1) (cid:27) , (4.4)provided d = 0 , − , − , . . . . Then we have the following theorem: Theorem 2
For d , d = 0 , − , − , . . . , the following result holds true: e π/ (cid:18) d + 316 d + 2740 (cid:19) + e − π/ (cid:18) d − d + 1140 (cid:19) = F (cid:18) i, − i, d + 1 , d ; 12 (cid:19) + √ F (cid:18) + i, − i, d + 1 , d ; 12 (cid:19) . (4.5) Proof.
In the first F series use (4.3) and in the second F series use (4.4) together withstandard properties of the gamma function. ✷ Corollary 4
If in (4.5) we take d = 1 / (7 n − and d = 15 / (24 n − for positive integer n then we find ne π/ = F (cid:18) i, − i, n − n − , n − ; 12 (cid:19) + √ F (cid:18) + i, − i, n +124 n − , n − ; 12 (cid:19) . (4.6)When n = 1 we recover (4.1). For n = 2 , e π/ = F (cid:18) i, − i, , ; 12 (cid:19) + √ F (cid:18) + i, − i, , ; 12 (cid:19) (4.7)and 3 e π/ = F (cid:18) i, − i, , ; 12 (cid:19) + √ F (cid:18) + i, − i, , ; 12 (cid:19) . (4.8)Similarly other results can be obtained.To conclude we note that an obvious extension of (1.1) is e πλ = F ( iλ, − iλ ; ; 1) + 2 λ F ( + iλ, − iλ ; ; 1) , (4.9)where λ is real. This yields the alternative expression for e ± π/ given by e ± π/ = F ( i, − i ; ; 1) ± F ( + i, − i ; ; 1) . (4.10)In addition, if we choose λ = √ √ √
67 and √