Parameterized Telescoping Proves Algebraic Independence of Sums

Usually creative telescoping is used to derive recurrences for sums. In this article we show that the non-existence of a creative telescoping solution, and more generally, of a parameterized telescoping solution, proves algebraic independence of certain types of sums. Combining this fact with summation-theory shows transcendence of whole classes of sums. Parameterized telescoping Given f1(k), . . . , fd(k) over a fielda K; find constants c1, . . . , cd ∈ K and g(k) such that g(k + 1)− g(k) = c1 f1(k) + · · ·+ cd fd(k). (1) If one succeeds, one gets the sum-relation g(n + 1)− g(0) = c1 n ∑ k=0 f1(k) + · · ·+ cd n ∑ k=0 fd(k). Telescoping: Restrict to d = 1. Zeilberger’s creative telescoping: Take a bivariate sequence f (m, k) and set fi(k) := f (m + i − 1, k) in (1). In the summation package Sigma [Sch07b] parameterized telescoping can be solved in Karr’s ΠΣ-fields: the fi(k) can be indefinite nested sums and products. a All rings and fields contain Q. ΠΣ∗-extensions and sequences Example. Let F := Q(m)(k)(h)(b) be a rational function field and define the field automorphism σ by σ(c) = c ∀c ∈ Q(m), σ(k) = k + 1, σ(h) = h + 1 k + 1 , Hk+1 = Hk + 1 k + 1 , σ(b) = m − k k + 1 b, ( m k + 1 ) = m − k k + 1 ( m k ) . (F,σ) is a difference field, more precisely, a ΠΣ∗-field. Difference rings and fields: A difference field (F,σ) is a ring (resp. field) F together with a ring (resp. field) automorphism σ : F → F; the constant ring (resp. constant field) is given by constσF := { f ∈ F|σ( f ) = f}. ΠΣ∗-field: A difference field (F(t),σ′) is a Σ∗-extension (resp. Π-extension) of a difference field (F,σ) :⇔ 1 t is transcendental over F, 2 σ′( f ) = σ( f ) for all f ∈ F, 3 σ′(t) = t + f (resp. σ′(t) = f t) for some f ∈ F∗, 4 the constant field remains unchanged, i.e., constσ′F(t) = constσF. A ΠΣ∗-extension is either a Πor a Σ∗-extension. (F(t1) . . . (te),σ) is a ΠΣ∗-extension (resp. Σ∗-extension or Π-extension) if it is a tower of such extensions. (F(t1) . . . (te),σ) is a ΠΣ∗-field over F, if F = constσF. Example. Each of the extensions k, h, b forms a ΠΣ∗-extension over the field below. In particular, constσQ(m)(k)(h)(b) = Q(m). Ring of sequences: The set of sequences over a field K is denoted by S(K) := {(an)n≥0|ai ∈ K}; we identity two sequences if they agree from a certain point on. The difference ring (S(K), S) with the shift operation (ring automorphism) S : 〈a0, a1, a2, . . . 〉 7→ 〈a1, a2, a3, . . . 〉 is called the ring of sequences. Goal: Embed, e.g., Q(m)(k)[h, b] into (S(Q(m)), S). Embedding example We construct step by an embedding (Q(m)(k)[h][b],σ) into (S(Q(m)), S). • Start with τ0 : Q(m) → S(Q(m)) where τ0(c) = 〈c, c, c, . . . 〉 ∀c ∈ Q(m). • Next, define the ring homomorphism τ1 : Q(m)(k) → S(Q(m)) with τ1( p q) = 〈F(k)〉k≥0 where F(k) =    0 q(k) = 0 p(k) q(k) q(k) 6= 0. . Note that τ1(σ( f )) = S(τ1( f )), ∀ f ∈ Q(m)(k). τ1 is injective: Since p(k), q(k) have only finitely many roots, τ1( p q) = 0 if and only if p(k) q(k) = 0. Hence τ1 is injective. • Define the ring homomorphism τ2 : Q(m)(k)[h] → S(Q(m)) with τ2(h) = 〈Hn〉n≥0 and τ2( d ∑ i=0 fih ) = d ∑ i=0 τ1( fi)τ2(h). τ2 is injective: If not, take f = ∑i=0 fih ∈ Q(m)(k)[h]∗ with deg( f ) = d minimal such that τ2( f ) = 0. Note that f / ∈ Q(m)(k) (otherwise, 0 = τ2( f ) = τ1( f ); since τ1 is injective, f = 0). Define g := σ( fd) f − fdσ( f ) ∈ Q(m)(k)[h]. Note: deg(g) < d by construction. Moreover, τ2(g) = τ1(σ( fn)) τ2( f ) } {{ } =0 −τ1( fn) τ2(σ( f )) } {{ } S(τ2( f ))=S(0)=0 = 0. By the minimality of deg( f ), g = 0, i.e., σ( fd) f − fdσ( f ) = 0. Equivalently, σ( f ) f = σ( fd) fd ∈ Q(m)(k). With f / ∈ Q(m)(k) this contradicts to [Kar81]. • To this end, take the ring homomorphism τ3 : Q(m)(k)[h][b] → S(Q(m)) with τ3(b) = 〈 m n ) 〉n≥0 and τ3( d ∑ i=0 fib ) = d ∑ i=0 τ2( fi)τ3(b). By similar arguments, τ3 is injective. Result 1: The embedding into the ring of sequences A generalized d’Alembertian extension (F(t1) . . . (te),σ) of (F,σ) is a ΠΣ∗-extension such that for all 1 ≤ i ≤ e, σ(ti)− ti ∈ F[t1, . . . , ti−1] or σ(ti)/ti ∈ F; note that the ti are transcendental and K := constσF(t1) . . . (te) = constσF. Embedding: Suppose that (F,σ) describes the rational case (F = K(k) with σ(k) = k + 1), the q-rational case or the mixed case. Then there is an injective ring homomorphism τ : F[t1 . . . , te] → S(K) with τ(c) = 〈c, c, c, . . . 〉 ∀c ∈ K such that the following diagram commutes: F[t1 . . . , te] τ // σ S(K)