Special values of the Gamma function at CM points M. Ram Murty & Chester Weatherby The Ramanujan Journal An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan ISSN 1382-4090 Ramanujan J DOI 10.1007/s11139-013-9531-x 1 23 Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media New York. This e-offprint is for personal use only and shall not be self- archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Ramanujan J DOI 10.1007/s11139-013-9531-x Special values of the Gamma function at CM points M. Ram Murty · Chester Weatherby Received: 10 November 2011 / Accepted: 15 October 2013 © Springer Science+Business Media New York 2014 Abstract Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z)when Q(z) is an imaginary quadratic field. We also study special values of the digamma function, ψ(z), and the polygamma functions, ψt (z). As part of our analysis we will see that certain infinite products ∞ A(n) nt n=1 can be evaluated explicitly and are transcendental for A(z) ∈ Q[z] with degree t and roots from an imaginary quadratic field. Special cases of these products were studied by Ramanujan. Additionally, we explore the implications that some conjectures of Gel’fond and Schneider have on these values and products. Keywords Transcendence · Gamma function · Digamma function · Polygamma functions · Infinite products · Gel’fond–Schneider conjecture Mathematics Subject Classification Primary 11J91 · Secondary 11J81 M.R. Murty’s Research partially supported by a Natural Sciences and Engineering Research Council (NSERC) grant. M.R. Murty Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6, Canada e-mail: [email protected] C. Weatherby (B) Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada e-mail: [email protected] Author's personal copy M.R. Murty, C. Weatherby 1 Introduction The Gamma function defined for (z) > 0as ∞ Γ(z):= e−t tz−1 dt 0 is well-studied, though√ much is unknown about the transcendence of Γ(z).Itis known that Γ(1/2) = π is transcendental since π is transcendental. In 1984, Chud- novsky [2] showed that Γ(1/3), Γ (2/3), Γ (1/4), Γ (3/4), Γ (1/6), and Γ(5/6) are transcendental, along with any integer translate of these fractions, by showing that firstly, π and Γ(1/3) are algebraically independent, and secondly, π and Γ(1/4) are algebraically independent. One gets the transcendence of Γ(1/6) by relating it to Γ(1/3) via the duplication formula − √ Γ(z)Γ(z+ 1/2) = 21 2z πΓ(2z). √ For z = 1/6, one can show that Γ(1/6) = 2−1/3 3/πΓ 2(1/3). The remaining values Γ(2/3), Γ (3/4), and Γ(5/6) are seen to be transcendental by the reflection formula π Γ(z)Γ(1 − z) = . sin(πz) It is unknown whether or not Γ(1/5) is algebraic or transcendental, though it is con- jectured to be transcendental. In 1941, Schneider [10] proved that the beta function Γ(a)Γ(b) B(a,b) := Γ(a+ b) is transcendental whenever a,b,a + b ∈ Q \ Z. By choosing a = b, we see that Schneider’s theorem implies that Γ(a) or Γ(2a) is transcendental for any a,2a ∈ Q \ Z. In particular, Γ(1/5) or Γ(2/5) is transcendental. More recently, Nesterenko [7] proved the following result which will be useful for our purposes. Theorem (Nesterenko) For any imaginary quadratic field with discriminant −D and character ε, the numbers √ D−1 π, eπ D, Γ(a/D)ε(a) a=1 are algebraically independent. = −D Here the character ε(n)√ ( n ) is the Kronecker–Jacobi symbol. This shows π D that in general, π and e are algebraically independent.√ Additionally, if we have D = 3, then we obtain the algebraic independence of π,eπ 3,Γ(1/3), and if D = 4, we see that π,eπ ,Γ(1/4) are algebraically independent. Author's personal copy Special values of the Gamma function at CM points In this article, we explore properties of the Gamma function at certain CM points (that is, z ∈ C such that Q(z) is an imaginary quadratic field). We show that all√ values Γ(α)√ and Γ(α+ 1/2) are transcendental for α/∈ Q of the form k + ia D/q ∈ Q( −D) for integers k,a,q. In particular, we will evaluate |Γ(ia/q)|2 explicitly for a,q ∈ Z and (a, q) = 1. Applying Nesterenko’s theorem allows us to deduce that Γ(ia/q)is always transcendental. The digamma function ψ(z) is the logarithmic derivative of Γ(z) while the polygamma functions ψt (z) are defined as the tth derivatives of ψ(z) with ψ0(z) = ψ(z). Transcendence of these values at rational values has been studied by Bund- schuh [1], the first author with Saradha [4, 5], the second author [13], among others. Here we focus on values of ψt (z) at irrational values. As a result of the analysis of Γ(z), we also obtain a method for evaluating certain infinite products ∞ A(n) (1) nt n=1 for A(z) ∈ Q[z] monic with degree t. We are able to relate these products to values of Γ(z)and derive results from there by specifying the roots of A(z). We point out that some of these results on infinite products of rational functions were known to Ramanujan [8] 100 years ago. At the time, many values which ap- peared in his calculations were not known to be transcendental. Some of the results in [8] are now implied by a more general method shown here with the addition of showing transcendence of these products. Indeed, in [8], we find some elegant for- mulas of the following kind: ∞ √ 2α 3 Γ(1 + α)3 sinh(πα 3) 1 + = √ α + n + n=1 Γ(1 3α)πα 3 and ∞ √ 2α + 1 3 Γ(1 + α)3 cosh(π( 1 + α) 3) 1 + = 2 . α + n Γ(2 + 3α)π n=1 If we put α = 0 in the second formula, we deduce the strikingly beautiful evaluation ∞ √ 1 cosh(π 3/2) 1 + = , (2) n3 π n=1 which is transcendental by Nesterenko’s theorem. One would expect these products to be transcendental for every rational α, but this deduction can only be made (at present) for a limited number of rational values of α like α = 1/3, 1/4or1/6. Analogous products appear in Ramanujan’s notebooks (see in particular Chaps. 13 and 14) and one finds formulas like (see [9]) ∞ x 2 |Γ(α)|2 1 + = . α + n |Γ(α+ ix)|2 n=1 These formulas will be special cases of our results. Author's personal copy M.R. Murty, C. Weatherby In the final section, we explore implications of conjectures of Schneider and Gel’fond on the algebraic independence of certain algebraic powers of algebraic num- bers. In some cases, we are able to prove conditional results on the transcendence of values of Γ(z),ψt (z) as well as infinite products (1). 2 Transcendental values of Γ(z)and infinite products The Gamma function can be written as an infinite product ∞ 1 z − = eγzz 1 + e z/n, (3) Γ(z) n n=1 where γ is Euler’s constant. The sine function has infinite product ∞ sin(πz) z2 = 1 − . (4) πz n2 n=1 Putting (3) and (4) together gives the reflection formula π Γ(z)Γ(1 − z) = . (5) sin(πz) Also recall the functional equation Γ(z+ 1) = zΓ (z), (6) which can be extended for a positive integer n to Γ(z) Γ(z+ n) = (z)nΓ(z) and Γ(z− n) = , (7) (z − n)n where we use the rising factorial or Pochhammer symbol (z)n := z(z + 1) ···(z + n − 1). From these elementary observations, we are able characterize certain special values of the Gamma function. √ √ = + ai D ∈ Z + Q − \ Z Theorem 1 Let α k q ( D) for a positive integer D. For k>0, − 2 1 − 2 π(1 α)2k−1 1 π( α)2k Γ(α) = and Γ α + = 2 , sin(πα) 2 cos(πα) while for k ≤ 0, 2 2 π 1 π Γ(α) = and Γ α + = . sin(πα)(α) | |+ 2 1 + 2 k 1 cos(πα)( 2 α)2|k| | |2 | + 1 |2 + 1 All of Γ(α) , Γ(α 2 ) , Γ(α), and Γ(α 2 ) are transcendental. Author's personal copy Special values of the Gamma function at CM points Proof Note that it is easily seen that Γ(z) = Γ(z) for any z ∈ C. Additionally, we shift values as in Eq. (7) so that for any non-integral algebraic number z, Γ(±n + z) is a nonzero algebraic multiple of Γ(z). Using these two observations as well as the reflection formula (5), for k positive we have that − 2 π(1 α)2k−1 Γ(α) = Γ(α)Γ(α) = (1 − α) k− Γ(α)Γ(1 − α) = 2 1 sin(πα) and 2 1 1 1 Γ α + = Γ α + Γ α + 2 2 2 1 1 1 1 π( − α)2k = − α Γ α + Γ 1 − α + = 2 .