Automatic Proofs of Identities: Beyond A=B

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1 FPSAC, Linz, July 20, 2009 Joint work with F. Chyzak and M. Kauers 1 / 28

2 I Introduction 2 / 28

Examples of Identities: Definite Sums, q-sums, Integrals n k=0 + 0 1 2πi n=0 ( ) H n (x)h n (y) un exp 4u(xy u(x 2 +y 2 )) n!

3 Examples of Identities: Definite Sums, q-sums, Integrals n k= πi n=0 ( ) H n (x)h n (y) un exp 4u(xy u(x 2 +y 2 )) n! = 1 4u 2 1 4u 2 q k2 (q; q) k (q; q) n k = n k= n ( 1) k q (5k2 k)/2 (q; q) n k (q; q) n+k xj 1 (ax)i 1 (ax)y 0 (x)k 0 (x) dx = ln(1 a4 ) 2πa 2 (1 + 2xy + 4y 2 ) exp ( 4x 2 y 2 1+4y 2 ) y n+1 (1 + 4y 2 ) 3 2 dy = H n(x) n/2! [Mehler1866] [Andrews1974] [GlasserMontaldi1994] [Doetsch1930] + multiple sums/integrals & many, many more in, e.g., 3 / 28

4 Examples of Non- Holonomic Identities 0 0 n k=0 ( ) n i(k + i) k 1 (n k + j) n k = (n + i + j) n k n ( ) { } n k ( 1) m k n + 1 n k! = m k k + 1 m k=0 m k=0 ( ) m B n+k = ( 1) m+n k n k=0 ( ) n B m+k k x k 1 ζ(n, α + βx) dx = β k B(k, n k)ζ(n k, α) x α 1 Li n ( xy) dx = π( α)n y α 0 sin(απ) x s 1 πy s Γ(s) exp(xy)γ(a, xy) dx = sin((a + s)π) Γ(1 a) [Abel1826] [Frobenius1910] [Gessel2003] 4 / 28

Computer Algebra Algorithms Examples: Aim Prove these identities automatically (fast?); Compute the rhs given the lhs; Explain why these identities exist.

5 Computer Algebra Algorithms Examples: Aim Prove these identities automatically (fast?); Compute the rhs given the lhs; Explain why these identities exist. 1st slide: Zeilberger s algorithm and variants (many people) 2nd slide (1st 3): Majewicz, Kauers, Chen & Sun; last 3: new generalization of previous ones. Ideas Confinement in finite dimension + Creative telescoping. 5 / 28

6 II Confinement in Finite Dimension and Closure Properties Idea: confine a function and all its derivatives/shifts/... 6 / 28

More Recent: same for linear differential or recurrence equations.

7 Finite Dimension and Special Functions Classical: polynomials represent their roots better than radicals. Algorithms: Euclidean division and algorithm, Gröbner bases. More Recent: same for linear differential or recurrence equations. Algorithms: non-commutative analogues & gen. func. About 25% of Sloane s encyclopedia, 60% of Abramowitz & Stegun. eqn+ini. cond.=data structure 7 / 28

8 First Proof: Mehler s Identity by Confinement n=0 ( ) H n (x)h n (y) un exp 4u(xy u(x 2 +y 2 )) n! = 1 4u 2 1 4u 2 1 Definition of Hermite polynomials H n (t): recurrence of order 2 vector space of dimension 2 over Q(t, n); 2 Product: vector space over Q(x, y, n) generated by H n (x)h n (y) n!, H n+1(x)h n (y) n!, H n(x)h n+1 (y), H n+1(x)h n+1 (y) n! n! recurrence of order at most 4; (confinement) 3 Translate into differential equation (and solve). 8 / 28

9 I. Definition O R 1 := {H(n C 2 ) =(K2 n K 2 ) H(n ) C 2 H(n C 1 ) x, H(0 ) = 1, H(1 ) = 2 x } : O R 2 := subs0h = H 2, x = y, R 1 1; II. Product O R 3 := gfun :- poltorec0 H(n )$ H 2 (n )$ v (n ), 2 R 1, R 2, {v (n C 1 ) $ (n C 1 ) = v (n ), v (1 ) = 1 } 3, 2 H(n ), H 2 (n ), v (n ) 3, c (n ) 1; O gfun :- rectodiffeq0 R 3, c (n ), f (u ) 1; u 3 K 16 u 2 y x K 4 u C 8 u y 2 C 8 u x 2 K 4 x y1 f (u ) C 0 16 u 4 K 8 u 2 C 1 1 Z d \ du f (u ) ] _, f (0 ) = 1 5 O dsolve(%, f (u ) ); O R 2 := 4 H 2 (0 ) = 1, H 2 (n C 2 ) = (K2 n K 2 ) H 2 (n ) C 2 H 2 (n C 1 ) y, H 2 (1 ) = 2 y5 R 3 := 4c (0 ) = 1, c (1 ) = 4 x y, c (2 ) = 8 x 2 y 2 C 2 K 4 y 2 K 4 x 2, c (3 ) = 32 3 x3 y 3 C 24 x y K 16 x y 3 K 16 x 3 y, (16 n C 16 ) c (n ) K 16 x y c (n C 1 ) C 0 K8 n K 20 C 8 y 2 C 8 x 2 1 c (n C 2 ) K 4 x c (n C 3 ) y C (n C 4 ) c (n C 4 )5 III. Differential Equation Z [ K4 x y u C y 2 C x 2 ] ^ \ (2 u K 1 ) (2 u C 1 ) _ I e f (u ) = e 0 Ky 2 K x u C 1 2 u K 1 9 / 28

10 Second Proof: Contiguity of Hypergeometric Series F (a, b; c; z) = (a) n (b) n z n, (c) n n! }{{} n=0 u a,n (x) n := x(x +1) (x +n 1). u a,n+1 (a + n)(b + n) = u a,n (c + n)(n + 1) z(1 z)f + (c (a + b + 1)z)F abf = 0, u a+1,n = n u a,n a + 1 S a F (a, b, c; z) := F (a + 1, b; c; z) = z a F + F. Notation: Differential and Shift Operators (a m (z)d m z + + a 0 (z)) F = a m (z)f (m) (z) + + a 0 (z)f (z) ( bp (k)s p k + + b 0(k) ) u k = b p (k)u k+p + + b 0 (k)u k. : any of S m, D x, q-shift, / 28

11 Second Proof: Contiguity of Hypergeometric Series F (a, b; c; z) = (a) n (b) n z n, (c) n n! }{{} u a,n n=0 (x) n := x(x +1) (x +n 1). u a,n+1 (a + n)(b + n) = u a,n (c + n)(n + 1) z(1 z)f + (c (a + b + 1)z)F abf = 0, u a+1,n = n u a,n a + 1 S a F (a, b, c; z) := F (a + 1, b; c; z) = z a F + F. Gauss 1812: contiguity relation. dim=2 S 2 a F, S a F, F linearly dependent: (Coordinates in Q(a, b, c, z).) S a D z 10 / 28

12 Gröbner Bases: Generalize Euclidean Division and Gcd 1 Monomial ordering: total order on the monomials, compatible with product, 1 minimal. 2 Gröbner basis of a (left) ideal I wrt : generators of I at the corners of its stairs. 3 Quotient modi: vector basis below the stairs (Vect{ α f }). 4 Reduction of P mod I: Unique remainder written on this basis. An access to (finite dimensional) vector spaces 11 / 28

13 Hilbert Dimension: a Handle on Infinite Dimension M s (I) := Vect{m m is below the stairs and of total degree s} Definition: Hilbert Dimension δ(i) dim M s (I) = O ( s δ(i)). Finite measure of infinite-dimensional vector-spaces. Can be obtained from a Gröbner basis. Definition (annihilator and -finiteness) Annf := {P P f = 0} f is -finite δ(ann f ) = 0 linear dim. of quotient is finite. 12 / 28

14 Examples Introduction Confinement Creative Telescoping Algorithms Conclusion Binomial coeffs ( n k) wrt Sn, S k ; Hypergeometric sequences: Bessel J ν (x) wrt S ν, D x ; Orthogonal pols wrt S n, D x : δ(i) = 0, dim S/I = 1 Stirling nbs wrt S n, S k δ(i) = 0, dim S/I = 2 Abel type wrt S m, S k, S r, S s hgm(m, k)(k + r) k (m k + s) m k r k+r : δ(i) = 1, dim S/I = δ(i) = 2 in space of dim / 28

15 Closure Properties + Proposition δ(ann(f + g)) max(δ(ann f ), δ(ann g)), δ(ann(fg)) δ(ann f ) + δ(ann g), δ(ann( f )) δ(ann f ). Algorithms by linear algebra (Gröbner bases as Input/Output). 14 / 28

16 III Creative Telescoping Closure under and Input: GB (Ann f ) Output: GB ( Ann f ) or GB (Ann f ) (or subideals). 15 / 28

17 Summation by Creative Telescoping Goal: evaluate U n := n k=0 GIVEN Pascal s triangle rule: ( ) ( ) ( ) ( n + 1 n n n = + = 2 k k k 1 k summing over k gives U n+1 = 2U n. ( ) n to 2 n. k ) + The initial condition U 0 = 1 concludes the proof. ( ) n k 1 ( ) n, k 16 / 28

18 Creative Telescoping (Zeilberger 1990) U n = b u n,k =? k=a GIVEN A(n, S n ) and B(n, k, S n, S k ) such that (A(n, S n ) k B(n, k, S n, S k )) u n,k = 0, then the sum telescopes, leading to A(n, S n ) U n = [B(n, k, S n, S k ) u n,k ] k=b+1 k=a Adapts easily to U(z) = b u k (z). k=a often = / 28

19 Creative Telescoping (Zeilberger 1990) U(z) = b a u(z, t) dt =? GIVEN A(x, D x ) and B(x, y, D x, D y ) such that (A(z, D z ) D t B(z, t, D z, D t )) u(z, t) = 0, then the integral telescopes, leading to A(z, D z ) U(z) = [B(z, t, D z, D t ) u(z, t)] t=b often t=a = 0. Then I come along and try differentating under the integral sign, and often it worked. So I got a great reputation for doing integrals. Richard P. Feynman 1985 Creative telescoping= differentiation under integral+ integration by parts 17 / 28

20 Diff. under + Integration by Parts Algorithm? Ex.: 1 0 cos zt dt = π 1 t 2 2 J 0(z), (zj 0 + J 0 + zj }{{} 0 = 0, J 0 (0) = 1). A(z,D z ) J 0 cos zt Ann A(z, D t 2 1 z) D 1 t 2 }{{} t D z } t{{} no t, D t anything Specification for a Creative Telescoping Algorithm Input: generators of (a subideal of) Ann f ; Output: A free of t, t, certificate B, such that A t B Ann f. Definition (Telescoping of I wrt t) T t (I) := (I + t Q(z, t) z, t ) Q(z) z. + variants for multiple sum/int. 18 / 28

21 IV Algorithms 19 / 28

22 Example: Rediscovering Pascal s Triangle Rule 1 Gröbner basis for Ann ( n k) : S n n + 1 n + 1 k 1, 2 Reduce all monomials of degree s = 2: S 2 n (n + 2)(n + 1) (n + 2 k)(n + 1 k) 1, S 2 k S k n k k over Q(n, k) (n k 1)(n k) 1, S n S k n + 1 (k + 2)(k + 1) k Common denominator: D 2 = (k + 1)(k + 2)(n + 1 k)(n + 2 k). D 2, D 2 S n, D 2 S k, D 2 S 2 n, D 2 S 2 k, D 2S n S k confined in Vect Q(n) (1, k1, k 2 1, k 3 1, k 4 1) ( ) n D 2 (S n S k S k 1) Ann. k 4 This has to happen for some degree: deg D s = O(s). over Q(n) 20 / 28

23 More Examples Proper hypergeometric [Wilf & Zeilberger 1992]: Q(n, k)ξ k u i=1 (a in + b i k + c i )! v i=1 (u in + v i k + w i )!, Q polynomial, ξ C, a i, b i, u i, v i integers: essentially the same situation. 1 n 2 + k 2 : confinement in a space of dimension O(s2 ), no elimination of k succeeds. f = a(z, t 1,..., t r ) b(z, t 1,..., t r ) : D s = b s, confinement in a space of dimension O(s 1 ) over Q(z), elimination of t 1,..., t r has to succeed. Base case of the proof that D-finite functions are holonomic. 21 / 28

24 Polynomial Growth and Creative Telescoping when δ > 0 Definition (Polynomial Growth p) There exists a sequence of polynomials P s (z 1,..., z k, t), s.t. a + b s P s a 1 z 1 a k z k b t pol of degree O(s p ) in t. Theorem (Chyzak, Kauers & Salvy 2009) δ(t t (I)) max(δ(i) + p 1, 0). Corollary (Sufficient Condition for Creative Telescoping) δ(i) + p 1 < k identities exist for the sum/int wrt t. Proof. Same as above. Also an algorithm. 22 / 28

25 Non- -Finite Examples (both with p = 1) Stirling: δ = 1 for 3 vars, e.g., Frobenius: n ( n k ( 1) m k k! m k k=0 ) { } n + 1 = k + 1 n. m Abel type: δ = 2 for 4 vars, e.g., Abel: n k=0 ( ) n i(k +i) k 1 (n k +j) n k = (n+i +j) n. k 23 / 28

26 Algorithm I. Sister Celine Style Polynomial growth + linear algebra J := I Q(z) z, t. Algorithm: Eliminate t For increasing s until δ(j ) bound, 1 Reduce all a z b t with a + b s; 2 Normalize to a common denominator; 3 Set up a linear system to cancel the positive powers of t; 4 If a solution is found, it has the form A(z, z ) + t B(z, z, t ). Return it. This computes in not in ((I Q(z) z, t ) + t Q(z) z, t ) Q(z) z, T t (I) := (I + t Q(z, t) z, t ) Q(z) z. 24 / 28

27 Algorithm II. Zeilberger Style Extended to δ > 0 Compute in T t (I) := (I + t Q(z, t) z, t ) Q(z) z Faster, more precise. 1 Hypergeometric case: Zeilberger 1990; t Algorithm (Zeilberger & variants) for s = 0, 1, 2,..., until found: 1 reduce A t B with A := η α (z) α, B := φ(z, t), α s for undetermined rational η α (z), φ(z, t). z 3 solve by an extended Gosper algorithm return the pairs (A, B). 25 / 28

28 Algorithm II. Zeilberger Style Extended to δ > 0 Compute in T t (I) := (I + t Q(z, t) z, t ) Q(z) z Faster, more precise. 1 Hypergeometric case: Zeilberger 1990; 2 -finite case (δ = 0): Chyzak 2000; Algorithm (Chyzak) t z2 for s = 0, 1, 2,..., until δ(j )= 0: 1 reduce A t B with A := φ β (z, t) β, η α (z) α, B := α α s β for undetermined rational η α (z), φ β (z, t). 2 extract coeffs of to form a linear system z1 of first order w.r.t. t 3 solve and set J to the ideal of the A s return the pairs (A, B). 25 / 28

29 Algorithm II. Zeilberger Style Extended to δ > 0 z1 Compute in T t (I) := (I + t Q(z, t) z, t ) Q(z) z Faster, more precise. 1 Hypergeometric case: Zeilberger 1990; 2 -finite case (δ = 0): Chyzak 2000; 3 Non- -finite: Algorithm (new) t z2 for s = 0, 1, 2,..., until δ(j ) bound: 1 reduce A t B with A := η α (z) α, B := α α s β M s(i) φ β (z, t) β, for undetermined rational η α (z), φ β (z, t). 2 extract coeffs of M s+1 (I) to form a linear system of first order w.r.t. t 3 solve and set J to the ideal of the A s return the pairs (A, B). 25 / 28

30 Final Example k ( n k ) { k l 1 Gröbner bases for ( n k), { k l } { } n k = m }, { } n k : m ( l + m l ) { } n m + l {(k n 1)S n +n+1, (k+1)s k +k n, S m 1, S l 1}, {S k S l (l+1)s l 1, S n 1, S m 2 Product by closure: {(k+1)(m+1)s k S l S m +(k n)s m +(1+k)S k S l +(1+l)(k n)s l S m, (1+k)S k S l S n 3 Creative telescoping: s = 1, 2 nothing; s = 3: system k(k + 1) A = S l S m S n (l+m+2)s l S m S m S l, B = k 2 1 n kn S (m + 1)k l+ k n 1 S ms l. 4 δ = 2 stop 26 / 28

31 V Conclusion 27 / 28

32 Conclusion Introduction Confinement Creative Telescoping Algorithms Conclusion Summary: Linear differential/recurrence equations as a data structure; Confinement in vector spaces + creative telescoping identities; Input dimension + polynomial growth output dimension. Also: Multiple summation/integration; Bounds identities + their size + complexity. Open questions: Replace polynomial growth by something intrinsic; Exploit symmetries; Compute all of T t (I); Structured Padé-Hermite approximants; Understand non-minimality. 28 / 28

The Dynamic Dictionary of Mathematical Functions

The Dynamic Dictionary of Mathematical Functions Bruno.Salvy@inria.fr AISC 2010. July 6, 2010 Joint work with: Alexandre Benoit, Alin Bostan, Frédéric Chyzak, Alexis Darrasse, Stefan Gerhold, Marc Mezzarobba