Euler’s factorial series and global relations Tapani Matala-aho1 and Wadim Zudilin2,3 1Matematiikka, PL 3000, 90014 Oulun yliopisto, Finland 2Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit, PO Box 9010, 6500 GL Nijmegen, The Netherlands 3School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia 7 March 2017 To the memory of Marc Huttner (1947–2015) Abstract k Using Pad´eapproximations to the series E(z) = ∞ k! ( z) , we address k=0 − arithmetic and analytical questions related to its valuesP in both p-adic and Archimedean valuations. 1 Introduction In his 1760 paper on divergent series [5], L. Euler introduced and studied the formal (hypergeometric) series ∞ E(z) := k!( z)k (1) − Xk=0 arXiv:1703.02633v1 [math.NT] 7 Mar 2017 (see also [1, 2] and, in particular, [8, Section 2.5]), which specializes at z = 1 to Wallis’ series ∞ W := ( 1)kk! , (2) − Xk=0 which has teased people’s imagination since the times of Euler. Notice that the series (1) is a perfectly convergent p-adic series in the disc z p 1 for all primes p, so that one can discuss the arithmetic properties of its values,| for| example≤ , at z = 1 (the Wallis case) and at z = 1. The irrationality of the latter specialization, − ∞ K = Kp := k! , (3) Xk=0 1 for any prime p is a folklore conjecture [12, p. 17] (see [11], also for a link of the problem to a combinatorial conjecture of H. Wilf). Already the expectation K = 1 (so that | p|p Kp is a p-adic unit) for all primes p, which is an equivalent form of Kurepa’s conjecture [7] from 1971, remains open; see [4] for the latest achievements in this direction. In what follows, we refer to the series in (1) as Euler’s factorial series and label it Ep(z) when treat it as the function in the p-adic domain for a given prime p. Theorem 1. Given ξ Z 0 , let be a subset of prime numbers such that ∈ \{ } P n 2 2 lim sup c n! n! p =0, where c = c(ξ; ) := 4 ξ ξ p. (4) n | | P | | | | →∞ pY pY ∈P ∈P Then either there exists a prime p for which Ep(ξ) is irrational, or there are two distinct primes p, q such that E∈( Pξ) = E (ξ) (while E (ξ), E (ξ) Q). ∈ P p 6 q p q ∈ Because n! =1/n! when the product is taken over all primes, condition (4) is p | |p clearly satisfiedQ for any subset whose complement in the set of all primes is finite. P This also suggests more exotic choices of . Furthermore, the conclusion of the theorem is contrasted with Euler’s sum P ∞ k k!= 1, · − Xk=0 n 1 which is valid in any p-adic valuation (and follows from − k k! = n! 1)—an k=0 · − example of what is called a global relation. P The idea behind the proof of Theorem 1 is to construct approximations pn/qn to the number ωp = Ep(ξ) in question, which do not depend on p and approximate the number considerably well for each p-adic valuation: q ω p = r for n =0, 1, 2,... ; n p − n n,p r 0 as n and there are infinitely many indices n for which r = 0 for | n,p|p → → ∞ n,p 6 at least one prime p . Assume that ωp = a/b, the same rational number, for all p . Then q a p∈b P Z 0 , so that 0 < q a p b 1, for infinitely many ∈ P n − n ∈ \{ } | n − n |p ≤ indices n and all primes p, hence 1= q a p b q a p b q a p b q a p b | n − n | | n − n |p ≤| n − n | | n − n |p Yp pY ∈P ( a + b ) max q , p br ≤ | | | | {| n| | n|} | n,p|p pY ∈P ( a + b ) max q , p r ≤ | | | | {| n| | n|} | n,p|p pY ∈P for those n. This means that the condition lim sup max qn , pn rn,p p =0 (5) n {| | | |} | | →∞ pY ∈P contradicts the latter estimate, thus making the -global linear relation ωp = a/b impossible. P 2 The result in Theorem 1 can be put in a general context of global relations for Euler-type series; the corresponding settings can be found in the paper [3]. We do not pursue this route here as our principal motivation is a sufficiently elementary arithmetic treatment of an analytical function that have some historical value. The rational approximations to E(z) we construct in Section 2 are Pad´eapproximations; in spite of being known for centuries, implicitly from the continued fraction for E(z) given by Euler himself [5] and explicitly from the work of T. J. Stieltjes [13], these Pad´e approximations remain a useful source for arithmetic and analytical investigations. In Section 3 we revisit Euler’s summation of Wallis’ series (2) using the approximations; we also complement the derivation by providing ‘Archimedean analogue(s)’ for the divergent series K = E( 1) from (3) — the case when the classical strategy does not − work. Our way of constructing the Pad´eapproximations is inspired by a related Pad´e construction of M. Hata and M. Huttner in [6]. The construction was a particular favourite of Marc Huttner who, for the span of his mathematical life, remained a passionate advocate of interplay between Picard–Fuchs linear differential equations and Pad´eapproximations. We dedicate this work to his memory. 2 Hypergeometric series and Pad´eapproximations Euler’s factorial series (1) is the particular a = 1 instance of the hypergeometric series ∞ k 2F0(a, 1 z)= (a) z . (6) | k Xk=0 Here and in what follows, we use the Pochhammer notation (a)k which is defined inductively by (a)0 = 1and(a)k+1 =(a+k)(a)k for k Z 0. Our Pad´eapproximations ≥ below are given more generally for the function (6). ∈ Theorem 2. For n, λ Z 0, take ∈ ≥ n i n i n ( 1) z − Bn,λ(z)= − . i (a) + Xi=0 i λ Then degz Bn,λ = n and for a polynomial An,λ(z) of degree degz An,λ n + λ 1 we have ≤ − B (z) 2F0(a, 1 z) A (z)= L (z), (7) n,λ | − n,λ n,λ where ordz=0 Ln,λ(z)=2n + λ. Explicitly, ∞ n + k n + k + a + λ 1 L (z)=( 1)nn! z2n+λ k! − zk. (8) n,λ − k k Xk=0 Proof. Relation (7) means that there is a ‘gap’ of length n in the power series expansion B (z) 2F0(a, 1 z)= A (z)+ L (z). n,λ | n,λ n,λ 3 n h Write Bn,λ(z)= h=0 bht and consider the series expansion of the product P ∞ l B (z) 2F0(a, 1 z)= r z , n,λ | l Xl=0 where rl = h+k=l bh(a)k; in particular, P n n i n (a)i+λ+m i n rl = ( 1) = ( 1) (a + i + λ)m − i (a) + − i Xi=0 i λ Xi=0 with m = l n λ for l > n + λ 1. To verify the desired ‘gap’ condition, − − − rn+λ = rn+λ+1 = = rn+λ+n 1 =0, (9) ··· − we introduce the shift operators N = N and ∆ = ∆ = N id defined on functions a a − f(a) by Nf(a)= f(a +1) and ∆f(a)= f(a + 1) f(a). − It follows from n n ∆ (a + λ)m =( 1) ( m)n(a + λ + n)m n for n Z 0 − − − ∈ ≥ that n n r = ( 1)i N i(a + λ) = (id N)n(a + λ) l − i m − m Xi=0 n =( ∆) (a + λ)m =( m)n(a + λ + n)m n, − − − which, in turn, implies (9) because of the vanishing of ( m) for m =0, 1,...,n 1. − n − The explicit expression for rl just found also gives n n + k n + k + a + λ 1 r2 + + =( n k) (a + λ + n) =( 1) n! k! − , n λ k − − n k − k k hence the closed form (8). We also have n+λ 1 n+λ 1 n − − l l i n (a)i+l n An,λ(z)= rlz = z ( 1) − . − i (a)i+λ Xl=0 Xl=0 Xi=0 i n l ≥ − This concludes our proof of the theorem. From now on we choose a = 1, λ = 0, change z to z and renormalize the − corresponding Pad´eapproximations produced by Theorem 2 by multiplying them by 4 ( 1)nn!: − n n 2 n n n! n i n i Q (z) := ( 1) n! B 0( z)= z − = i! z , n − n, − i i! i Xi=0 Xi=0 n 1 n − n n l n i n n!(i + l n)! P (z) := ( 1) n! A 0( z)=( 1) ( z) ( 1) − − n − n, − − − − i i! Xl=0 i=Xn l − n 1 l 2 − n =( 1)n ( z)l ( 1)ii!(l i)! − − − − i Xl=0 Xi=0 and 2 ∞ n 2 2n k n + k k R (z) := ( 1) n! L 0( z)= n! z ( 1) k! z . n − n, − − k Xk=0 In this notation, the Pad´eapproximation formula (7) may be rewritten as Q (z)E(z) P (z)= R (z) (10) n − n n for n Z 0. Observe that ∈ > 2 2n Q (z)P +1(z) Q +1(z)P (z)= n! z . (11) n n − n n Indeed, the standard Pad´ewalkabout proves the identity Q (z)P +1(z) Q +1(z)P (z)= Q +1(z)R (z) Q (z)R +1(z), n n − n n n n − n n in which the degree of the left-hand side is at most 2n while the order of the right-hand side is at least 2n.