Note on holonomic constants Errata: efficient accelero-summation of holonomic functions

JORIS VAN DER HOEVEN

February 27, 2021

There were several problems with the proofs of [2, Proposition 4.7(c) and (d)]. In this note, we present corrected proofs (in the case of [2, Proposition 4.7(c)], we slightly mod- ified the statement), as well as a theorem that the class 핂rhol regular singular holonomic constants is essentially the same as the class of 핂hol of ordinary holonomic constants. Until the very last subsection, we will assume that 핂=ℚalg is the field of algebraic num- bers. In the last subsection, we also discuss a few related questions and results from [1]; we are grateful to Marc Mezzarobba for this reference. Throughout our note, we will freely use notations and references from [2].

Other Errata for [2] • Remark 7.4 is wrong. In fact, thanks to optimizations by Marc Mezzarobba [4], the top two entries of the rightmost column of Table 7.1 can now be replaced by O(M(n)log2 n). • The last sentence of the first paragraph of Appendix A is slightly misleading; it ˜ should be “Indeed, for any N ∈ℕ, we may compute (sumN f )(z) using ...”; the N that we take is generally smaller than the maximal N allowed by summation up ˜ to the least term. In fact, if |log z| ≽ log n and size(z) = O(log z), then (sumN f )(z) can be computed in time O(M(n) log n) through a refined analysis of the binary splitting algorithm in this context. • In Theorem A.1, we need to require that size(z)=O(n). The proof in the case when size(z)≻log n is not completely trivial. It is given in [3, Proposition 7] for expedito- summation. If size(z)=O(log n), then the bound actually becomes O(M(n)log n), as noted above.

Notations Let ℒ hol and ℒ shol denote for the sets of monic L∈핂(z)[∂] whose coefficients are respec- ¯ rhol shol tively defined on 풟0,1 and 풟 0,1. Let ℒ be the set of L ∈ ℒ such that L is at worst regular singular at z = 1. We define ℱ hol, ℱ rhol, and ℱ shol to be the sets of solutions f ∈ 핂{{z}} to an equation Lf = 0, where L ∈ ℒ hol, L ∈ ℒ rhol, or L ∈ ℒ shol, respectively, and hol hol rhol such that limz→1 f (z) exists. We recall that 핂 = { f (1) : f ∈ ℱ }, 핂 = {limz→1 f (z) : rhol shol shol f ∈ℱ }, and 핂 ={limz→1 f (z) : f ∈ℱ }. It will be convenient to also introduce the variants ℒ hola, ℒ rhola, and ℒ shola of ℒ hol, ℒ rhol, and ℒ shol for which we allow L to be at most regular singular at z=0. For instance, hola ¯ ℒ consists of monic operators L∈핂(z)[∂] whose coefficients are defined on 풟 0,1 ∖{0} and such that L is at worst regular singular at z = 0. The counterparts ℱ hola, ℱ rhola, and ℱ shola are defined in a similar way as before; we still require analytic solutions f ∈핂{{z}} hola hola rhol rhola of Lf = 0 at z= 0. We again set 핂 = { f (1) : f ∈ℱ }, 핂 = {limz→1 f (z) : f ∈ℱ }, shola shola and 핂 = {limz→1 f (z): f ∈ℱ }.

1 2 NOTE ON HOLONOMIC CONSTANTS

Ring structure

PROPOSITION 1. 핂hol, 핂rhol, 핂shol, 핂hola, 핂rhola, and 핂shola are all subrings of ℂ.

Proof. This is proved in a similar way as Proposition 4.6(a). For instance, in order to see that 핂rhol is closed under multiplication, consider solutions f and g of Kf = 0 and Lg = 0 with initial conditions in 핂m resp. 핂n, where the coefficients of K and L are defined on 풟0,1, where K and L are regular singular at z=1, and such that the limits of f and g at z = 1 exist. Then Corollary 4.4 implies that K ⊠ L is defined on 풟0,1 and Corol- lary 4.5 implies that K ⊠ L is regular singular at z = 1. Consequently, limz→1 ( f g)(z) = rhol (limz→1 f (z)) (limz→1 g(z)) belongs to 핂 . □

This proposition also allows us to consider initial conditions in 핂hol instead of 핂 in many circumstances. For instance, by definition, the value of a f ∈ ℱ hol at ¯ hol a point in 풟0,1 ∩ 핂 lies in 핂 . Thanks to the proposition, this even holds for solutions f ∈ 핂hol{{z}} to an equation Lf = 0 with f ∈ ℒ hol. Indeed, given z ∈ 핂, we have F(z) = hol Δ0→z F(0); since Δ0→z and F(0) both have coefficients in 핂 , the same holds for F(z).

Regular singular transition matrices

LEMMA 2. Let L ∈ ℒ hola, 훼 ∈ 핂, and consider a solution f ∈ z훼i 핂{{z}}[log z] of the equation Lf = 0. Then f ∈z훼i ℱ hola[log z].

Proof. Without loss of generality, we may assume that 훼 = 0. Now write f = d 2πi d fd (log z) + ⋅ ⋅ ⋅ + f0 with f0, . . . , fd ∈핂{{z}}. Then f (z e ) = fd (log z + 2 π i) + ⋅ ⋅ ⋅ + f0 ∈ 핂{{z}}[log z][2πi] is also annihilated by L. Since 2πi is transcendental, each of the coef- ficients of f (ze2πi) as a in 2πi is again annihilated by L; these coefficients are

d fd, d fd log z + fd−1, . . ., fd (log z) + ⋅ ⋅ ⋅ + f0. It follows that

hola hola hola hola hola d fd ∈ℱ , fd−1 ∈ℱ +ℱ log z, . . ., f0 ∈ℱ + ⋅ ⋅ ⋅ +ℱ (log z) , whence f ∈ℱ hola[log z]. □

PROPOSITION 3. Let L be a linear of order n in 핂(z)[∂]. Then Δ훾 ∈ hola Matn(핂 ) for any regular singular broken-line path 훾 as in section 4.3.2.

Proof. In view of (4.6), it suffices to prove the result for paths of the form 휎휃 →휎 + z and for paths of the form 휎 + z → 휎휃. Without loss of generality we may assume that 휎 = 0. hola 핂 By what precedes, the entries of Δ0휃→z as functions in z are all in ℱ [z ][log z]. Now values of functions z훼 (log z)k with 훼∈ 핂 and k ∈ ℕ at points z ∈핂≠ are in 핂hol. Conse- ≠ hola quently, values of entries of Δ0휃→z at z ∈ 핂 are in 핂 . Let h1, . . . , hr be the canonical basis of solutions of Lf = 0 at the origin. Then we recall that

h1(z) ⋅ ⋅ ⋅ hr(z) ( ⋅ ⋅ ) Δ = ( ⋅ ⋅ ). 0휃→z ( ⋅ ⋅ ) ( (r−1) (r−1) ) ( h1 (z) ⋅ ⋅ ⋅ hr (z) ) JORIS VAN DER HOEVEN 3

The determinant W =Wh1, . . .,hr of this matrix satisfies the equation W′+Lr−1 W =0 and its −1 −1 −1 inverse W satisfies (W )′ − Lr−1 W = 0. Since L is at worst regular singular at z = 0, 훼 we have Lr−1 = z + Q, where 훼 ∈ 핂 and Q ∈핂(z) is analytic at z = 0. It follows that W = c−1 z−훼 e−∫Q and W −1 =cz훼 e∫Q for some c∈핂, where (∫Q)(0)=0 (here c∈핂 follows from (j) ≠ the fact that the coefficients of all hi are in 핂 as oscillatory transseries). Given z ∈ 핂 where W−1 is defined, it follows that W −1(z) ∈핂hol. Since 핂hola is a , it follows that the coefficients of −1 −1 Δz→0휃 = Δ0휃→z = W (z) adj(Δ0휃→z) are in 핂hola. □

COROLLARY 4. We have 핂shol ⊆핂hola.

shol shol shol Proof. Given c ∈핂 , let L∈ℒ and f ∈ℱ such that Lf =0 and c =limz→1 f (z). Let hola 휆1, . . . , 휆r ∈ 핂 be the entries of Δ0→1 F(0). Then f = 휆1 h1 + ⋅ ⋅ ⋅ + 휆r hr, where h1, . . . , hr is the canonical fundamental system of solutions of Lf = 0. Since limz→1 f (z) exists, we must have h = O(1) whenever 휆 ≠ 0 and f (1)= ∑ 휆 h (0)∈핂hola. □ i i i,휆i≠0 i i Alien operators Given an analytic function f defined on a neighbourhood of the origin on the Riemann surface of the , we define (∇ f )(z)= f (z)− f (ze−2πi). Setting Λ=(2πi)−1log z, the operator f (z)↦ f (ze−2πi) acts on ℂ{{z}}[Λ] by sending Λ to Λ− 1, whence ∇=1− e−∂Λ = 1 ∂Λ − /2 ∂Λ + ⋅ ⋅ ⋅. Given d ∈ ℕ, let ℂ{{z}}[Λ]

i;j i−1 i−1 sends ℂ{{z}}[Λ]

Eschewing regular singularities

THEOREM 5. We have 핂hol ⊆핂shol ⊆핂hola ⊆풳−1 핂hol. 4 NOTE ON HOLONOMIC CONSTANTS

Proof. The inclusion 핂hol ⊆ 핂shol is trivial and we already proved that 핂shol ⊆ 핂hola, so we focus on the remaining inclusion 핂hola ⊆풳−1 핂hol. Consider a monic L ∈ ℒ hola of order r. Then Lh = 0 has a canonical fundamental system of solutions

훼i 훼i j hi,j ∈z 핂{{z}}[log z]⩽j, hi,j ∼z (log z) , i= 1, . . ., ℓ, j=0, . . .,휈i −1.

In particular, we have r =휈1 + ⋅ ⋅ ⋅ + 휈ℓ. For each i∈{1, . . .,ℓ}, let

휈1 휈i−1 휈i+1 휈ℓ Πi ≔ ∇훼1 ⋅ ⋅ ⋅ ∇훼i−1 ∇훼i+1 ⋅ ⋅ ⋅ ∇훼ℓ , so that Πi f ∈핃훼i for any solution f of Lf =0. We also define

−2πi(훼i′−훼i) 휈i′ ui ≔ (1− e ) , i′≠i

훼i so that Πi f = ui f whenever f ∈z ℂ{{z}}. (r−1) hol Let f be a solution of Lf =0 with F(1)=( f (1),..., f (1))∈핂 and let 휆i,j ∈ℂ be such −1 hol that f =∑i,j 휆i,j hi,j. We need to show that 휆i,j ∈풳 핂 for all i and j. Given i∈{1,...,ℓ}, −1 hol let us show by induction on j that 휆i,j ∈ ui 핂 . To this effect, given j ∈ {0, . . . , 휈i − 1}, −1 hol −1 hol assume that 휆i,j+1, . . .,휆i,휈i−1 ∈ui 핂 , and let us show that 휆i,j ∈ui 핂 . j+1;휈i j+1;휈i Let g≔Ζ훼 f and let f훼 =휆i,0 hi,0 + ⋅⋅ ⋅ + 휆i,휈i−1 hi,휈i−1 and g훼 =Ζ훼 f훼 be the compo- 훼 nents of f and g in z ℂ{{z}}[log z]. By construction, g훼 has degree at most j in log z and the coefficient (g훼)j of degree j is of the form

(g훼)j = (휈i − j− 1)!휆i,j +cj+1 휆i,j+1 + ⋅ ⋅ ⋅ +c휈i−1 휆i,휈i−1 +o(1)

−1 for constants cj+1, . . .,c휈i−1 ∈핂[(2 πi) ] that can be computed explicitly. −훼i j We next consider the function 휑 = z Πi ∇훼 g. By construction, 휑 ∈ℂ{{z}} and

휑 = j!ui ((휈i − j−1)!휆i,j + cj+1 휆i,j+1 + ⋅ ⋅ ⋅ +c휈i−1 휆i,휈i−1 + o(1)). Moreover, both g and 휑 belong to ℋ, so the value of the contour 1 휑(z) 휑(0) = dz 2 πi |z|=1 z hol −1 hol actually lies in 핂 . By our assumption that 휆i,j+1, . . .,휆i,휈i−1 ∈ui 핂 , it follows that hol 휑(0) − j!ui (cj+1 휆i,j+1 + ⋅ ⋅ ⋅ + c휈i−1 휆i,휈i−1) ∈핂 , hol −1 hol whence ui 휆i,j ∈핂 . By induction on j, this shows that 휆i,j ∈ui 핂 , for all j. □

Remark 6. In the special case when ℓ = 1 or when 훼i − 훼j ∈ ℚ for all i, j, we note that the hol numbers ui are all in 핂. It follows that Δ0휃→z and Δz→0휃 have coefficients in 핂 .

Irregular singularities

PROPOSITION 7. Let L be a linear differential operator of order n in 핂(z)[∂]. Then Δ훾 ∈ shola Matn(핂 ) for any singular broken-line path 훾 as in section 4.3.2.

Proof. In view of (4.6), it suffices to prove the result for paths of the form 휎풌,휽 → 휎 + z or 휎 + z → 휎풌,휽. In fact, it suffices to consider paths of the form 휎풌,휽 → 휎 + z, by using a similar argument as in the regular singular case, based on the . Without loss of generality we may assume that 휎 =0. JORIS VAN DER HOEVEN 5

Now, as shown in detail in section 7.3, the matrix Δ0풌,휽→z can be expressed as a product of matrices whose entries are either evaluations of regular singular Borel trans- ˇ(m) ≠ forms f1 (a1) at a point a1 ∈핂 near the origin, or of the form

ei휃i∞ ˆ ˇ (m) 휑i(휁i) Kki,ki+1(휁i, ai+1)d휁i, (1) bi or ei휃p∞ −휁p/zp (m) 휑ˆ p(휁p)(e ) d휁i, (2) bp

shola where ai+1, bi, zp ∈핂, m ∈ℕ and 휑ˆ i is holonomic with initial conditions in 핂 . More- over, bi and bp may be chosen as large as desired. ˇ(m) ˆ ˇ Let us first consider the evaluations f1 (a1) near the origin. The functions f1 and f1 ˆ ˆ ˇ ˇ satisfy holonomic equations L1 f1 = 0 and L1 f1 that are regular singular at the origin. In particular, the transition matrices between 0 and a1 for these equations have entries hola ˜ 휎 r in 핂 . Thanks to the explicit formulas of (ℬ z1 z1 log z1)(휁1) in section 2.1, we see that ˆ (ℕ) f1 can be expressed as a 핂[훾 (핂)]-linear combination of the canonical solutions of Lˆ 1 at the origin, where 핂[훾 (ℕ)(핂)] is the smallest 핂-algebra that contains all constants of the form 훾 (m)(휎) with 훾(z) =1/Γ(z), m∈ℕ, and 휎 ∈핂. Now i 훾 (m)(z) = (−log (−t))m (−t)−z e−t dt, 2π ℋ for all m ∈ℕ, where ℋ is a Hankel contour from ∞ around 0 and then back to ∞. Such can be evaluated using the technique from section 6, so 핂[훾 (ℕ)(핂)] ⊆ 핂shol. Using the explicit formula for majors of functions of the form 휑(휁)휁 휎 logk 휁 in section 2.2, ˇ (ℕ) −2πi핂≠ −1 we next deduce that f1 is a 핂훾 (핂),1−e -linear combination of the canon- ˇ ical solutions of L1 at the origin. As we will see in the section below, we again have −2πi핂≠ −1 shol ˇ(m) shola ≠ 1 − e ⊆ 핂 . It follows that f1 (a1) ⊆ 핂 for a1 ∈ 핂 near the origin and all m∈ℕ. ˇ (m) −휁p/zp (m) By the results from section 4.2, the kernels Kki,ki+1(휁i, ai+1), (e ) and the inte- shola grands or (1) and (2) are all holonomic, with initial conditions in 핂 at bi. Note that the m-th derivatives are taken with respect to ai+1 and zp, so they amount to multiplying the integrands with a polynomial in 휁i or 휁p of degree m. Let us focus on the integrals of type (1); the integrals of type (2) are treated similarly. The function 휑ˆ i satisfies a holo- nomic equation (i.e. a monic linear differential equation with coefficients in 핂(휁i)) of ki/(ki−ki+1) which all solutions have a growth bounded by BeC|휁i| , for some fixed constant C>0 ˇ (m) and a constant B that depends on the solution. Likewise, as shown in section 7.1, Kki,ki+1(휁i, ki/(ki−ki+1) −C|휁i| ai+1) satisfies a holonomic equation of which all solutions are bounded by Be for a fixed constant C>0 that can be made arbitrarily large (by taking bi large). By Lemma ˆ ˇ (m) 4.3(b), it follows that the same holds for the integrand I(휁) ≔휑i(휁i) Kki,ki+1(휁i,ai+1). Given such a holonomic equation satisfied by I, consider the canonical fundamental basis h1, .. ., hs of solutions to this equation at 휁i =bi. For each j, the function hj has initial conditions in 핂 at bi and the integral

ei휃i∞ hj(휁i)d휁i, bi 6 NOTE ON HOLONOMIC CONSTANTS

converges. Taking bi sufficiently large and applying a change of variables of the form c −1 shol (휁i /bi) = (1 − 휉) , we see that the value of the integral lies in 핂 . Since I can be shola rewritten as a 핂 -linear combination of h1,...,hs, we conclude that (1) also takes a value in 핂shola. □

Invertible elements Given an integral domain R, let R× be its subgroup of invertible elements. An inter- esting question is to determine the sets (핂hol)×, (핂rhol)×, etc. Obviously, 핂≠ ⊆ (핂hol)× and e핂 ⊆(핂hol)×. We also know that πℤ ⊆(핂hol)×, since 1 1 π = 4arctan + arctan ∈ 핂hol 2 3 1 22 (4k)!(1103+ 26390k) = ∈ 핂hol, π 9801 (k!)4 3964k k∈ℕ 1 and π /2ℤ ⊆(핂shol)×, since 1 √휋 = Γ . 2 It would be interesting to know whether π훼 ∈ (핂shol)× for other rational numbers 1 훼∈ℚ ∖( /2 ℤ). From ∞ Γ(z) = xz−1 e−x dx 0 1 i = (−t)−z e−t dt, Γ(z) 2 π ℋ we deduce that Γ(핂∖ℤ) ⊆(핂shol)×, where ℋ is a Hankel contour from ∞ around 0 and then back to ∞. From the above facts and Euler's reflection formula π Γ(1 −z) Γ(z) = , sin (πz) we also deduce that sin (π (핂 ∖ ℤ)) ⊆ (핂shol)×. This is noteworthy, since sin z is 1 a well known example of a holonomic function whose inverse sin z is not holonomic. Apart from the invertible elements that directly follow from the above list of exam- ples, the author is not aware of any other invertible holonomic constants. In particular, the precise status of 풳 is unclear. From sin (π(핂∖ℤ))⊆핂shol, it follows that 풳−1 ⊆핂shol, whence 핂shol =핂shola. In combination with Proposition 7, this actually provides a correct proof of [2, Proposi- tion 4.7 (d)]. If 풳−1 ⊆핂hol, then this would also imply 핂hol=핂rhol=핂hola =핂rhola. It seems plausible though that 풳 ∩핂hol = {1} and 핂hol =핂rhol = 핂hola =핂rhola both hold.

Further comments For simplicity, we have assumed that 핂 = ℚalg is the of algebraic numbers, throughout our exposition. Most results go through without much change for arbitrary algebraically closed fields 핂. Only in the proof of Lemma 2, we used the assumption that 2 π i ∉ 핂; if 2 π i ∈ 핂, then the same conclusion can be obtained by induction on d, by applying the induction hypothesis on f (z e2πi)− f (z), when d>0. JORIS VAN DER HOEVEN 7

Another interesting direction of generalization would be to consider holonomic func- tions that are completely defined over ℚalg, but to consider values at points in larger π fields 핂. Such classes of constants contain numbers like ee , sin Γ2, etc. In [1], the authors consider values of so-called Siegel G-functions, which are a par- ticular type of Fuchsian holonomic functions. They prove an analogue of Theorem 5 in this setting. Their proof is significantly simpler, thanks to special properties of G-func- tions [1, Theorem 3], and based on similar arguments as our proof of Proposition 3. The paper [1] also contains several results about fraction fields of fields of values of G-func- tions. It would be interesting to investigate analogues of these results in our setting. alg Still in [1], the authors study the case when 핂⊆/ ℚ is an algebraic number field that is strictly contained in ℚalg. They showed that any real algebraic number can be obtained ¯ as the value at z = 1 of a G-function over ℚ that is defined on 풟0,1. In our setting, this immediately implies that ℚalg ⊆ℚhol[i], whence ℚalghol =ℚhol[i]. More generally, for any algebraic number field 핂, we obtain ℚalghol =핂hol[i] if 핂⊆ℝ and ℚalghol =핂hol if 핂⊆/ ℝ.

BIBLIOGRAPHY

[1] S. Fischler and T. Rivoal. On the values of G-functions. Technical Report http://arxiv.org/abs/ 1103.6022, Arxiv, 2011. [2] J. van der Hoeven. Efficient accelero-summation of holonomic functions. JSC, 42(4):389–428, 2007. [3] J. van der Hoeven. Uniformly fast evaluation of holonomic functions. Technical Report, HAL, 2016. http://hal.archives-ouvertes.fr/hal-01374898. [4] Marc Mezzarobba. Autour de l'évaluation numérique des fonctions D-finies. PhD thesis, École polytech- nique, 2011.