holonomy and combinatorics - simon fraser universitypeople.math.sfu.ca/~mmishna/pub/holtalk.pdf ·...
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Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 1/26
Holonomy and combinatoricsFrom automatic identities to systematic enumeration
Marni MishnaLaBRI, Université Bordeaux I
http://www.labri.fr/~mishna
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 2/26
A grab bag of problems
Prove Dixon’s identity:∑
k(−1)k(n+an+k
)(n+bb+k
)(a+ba+k
)= (n+a+b)!
n!a!b!
Calculate the dimensionof one representation ofthe symmetric group inanother
ζ(2, 2) = 1/2(ζ(2)2 − ζ(4)
)
Find a recursive formulafor the number of 4-
regular graphs of size n
Find a formula for standard
Young tableaux of size n
5 7
4 6 8
1 2 3
Enumerate plane parti-tions of shape 2× n
633321
522111
Prove q-Hypergeometric identities
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 3/26
Basic Idea
Set of easy identities
A set of recurrences forp(n, k)
A recurrence forφ(p(n, k))
Systems of differentialequations satisfied by f
and g
A differential equationfor ψ(f, g)
The recurrences and DEs are useful! They give sequences,can be solved, allow asymptotic analysis, ...
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 3/26
Basic Idea
Set of easy identities Elimination inoperator algebra
A set of recurrences forp(n, k)
A recurrence forφ(p(n, k))
Systems of differentialequations satisfied by f
and g
A differential equationfor ψ(f, g)
The recurrences and DEs are useful! They give sequences,can be solved, allow asymptotic analysis, ...
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 3/26
Basic Idea
Set of easy identities Elimination inoperator algebra Profound identity
A set of recurrences forp(n, k)
A recurrence forφ(p(n, k))
Systems of differentialequations satisfied by f
and g
A differential equationfor ψ(f, g)
The recurrences and DEs are useful! They give sequences,can be solved, allow asymptotic analysis, ...
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 3/26
Basic Idea
Set of easy identities Elimination inoperator algebra Profound identity
A set of recurrences forp(n, k)
A recurrence forφ(p(n, k))
Systems of differentialequations satisfied by f
and g
A differential equationfor ψ(f, g)
The recurrences and DEs are useful! They give sequences,can be solved, allow asymptotic analysis, ...
The Set Up
● A grab bag of problems
● Basic Idea
● Summary of how it works
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 3/26
Basic Idea
Set of easy identities Elimination inoperator algebra Profound identity
A set of recurrences forp(n, k)
A recurrence forφ(p(n, k))
Systems of differentialequations satisfied by f
and g
A differential equationfor ψ(f, g)
The recurrences and DEs are useful! They give sequences,can be solved, allow asymptotic analysis, ...
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 4/26
Summary of how it works
Dixon’s Identity 4-Regular Graphs
�
k(−1)k
�
n+a
n+k
� �
n+b
b+k
� �
a+b
a+k
�
=(n+a+b)!
n!a!b!
1
2
3 4
5
6
7
Operator Algebra Shift Differential
Closure Operation +, * 〈, 〉
“Easy” Identity recurrence for n! DE for ex
Elimination Answer is in a left ideal. Find a Gröbner basis for this ideal
Output Recurrences for hypergeo-metric sums
Diff eq. satisfied by gen-erating function
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 5/26
Creative telescoping
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 6/26
P-Recursive Functions
Def: A function f : N → R is P-recursive if it satisfies a finitelinear recurrence with polynomial coefficients.There exists polynomials γi such that:
γd(n)f(n+ d) + . . .+ γ1(n)f(n+ 1) + γ0(n)f(n) = 0
Ex: f(n) = n! is P-recursive: 1(n+ 1)!− (n+ 1)n! = 0Ex: Rational: p(x)/q(x) =
∑
n f(n)xn =⇒ γi = const.
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 7/26
Express recurrences as operators
Shift operator:
SnP (n, k) = P (n+ 1, k), SkP (n, k) = P (n, k + 1)
Define a (non-commutative) algebra:
Bn,k := C〈n, k, Sn, Sk〉
Snn = (n+ 1)Sn, Skk = (k + 1)Sk, all other variables commute
0 = P (n, k) + P (n, k + 1) + (3n2 + k)P (n+ 2, k + 1)
= P (n, k) + SkP (n, k) + (3n2 + k)S2nSkP (n, k)
= (1 + Sk + (3n2 + k)S2nSk) · P (n, k)
= ψ(n, k, Sn, Sk) · P
■ If ψ, ξ ∈ IP , then A(n, k, Sn, Sk)ψ +B(n, k, Sn, Sk)ξ ∈ IP .■ ψ is in the annihilating (left) ideal IP of P .
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 7/26
Express recurrences as operators
Shift operator:
SnP (n, k) = P (n+ 1, k), SkP (n, k) = P (n, k + 1)
Define a (non-commutative) algebra:
Bn,k := C〈n, k, Sn, Sk〉
Snn = (n+ 1)Sn, Skk = (k + 1)Sk, all other variables commute
0 = P (n, k) + P (n, k + 1) + (3n2 + k)P (n+ 2, k + 1)
= P (n, k) + SkP (n, k) + (3n2 + k)S2nSkP (n, k)
= (1 + Sk + (3n2 + k)S2nSk) · P (n, k)
= ψ(n, k, Sn, Sk) · P
■ If ψ, ξ ∈ IP , then A(n, k, Sn, Sk)ψ +B(n, k, Sn, Sk)ξ ∈ IP .■ ψ is in the annihilating (left) ideal IP of P .
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 7/26
Express recurrences as operators
Shift operator:
SnP (n, k) = P (n+ 1, k), SkP (n, k) = P (n, k + 1)
Define a (non-commutative) algebra:
Bn,k := C〈n, k, Sn, Sk〉
Snn = (n+ 1)Sn, Skk = (k + 1)Sk, all other variables commute
0 = P (n, k) + P (n, k + 1) + (3n2 + k)P (n+ 2, k + 1)
= P (n, k) + SkP (n, k) + (3n2 + k)S2nSkP (n, k)
= (1 + Sk + (3n2 + k)S2nSk) · P (n, k)
= ψ(n, k, Sn, Sk) · P
■ If ψ, ξ ∈ IP , then A(n, k, Sn, Sk)ψ +B(n, k, Sn, Sk)ξ ∈ IP .■ ψ is in the annihilating (left) ideal IP of P .
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 8/26
Making new identities using elimination
Recall from your youth:n equations, m unknowns variables + linear algebra =⇒eliminate some unknown variables.
3x+ 4y + z = 2 (1)
5x− 2y + 2z = 3 (2)
5 ∗ (1)− 3 ∗ (2) =⇒
26y − z = 1
We can do this in the setting of non-commutative algebras.(Ore algebra)■ Left ideals will have Gröbner basis (Galligo)■ Modified traditional tools work (Buchbergers + Euclidean
algorithm)
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
■ Easy identities: F (n+1,k)F (n,k) = n+1
n−k+1F (n,k+1)
F (n,k) = n−kk+1
■ Cross Multiply:
((n− k + 1)Sn − (n+ 1))F = 0 ♥
((k + 1)Sk − (n− k))F = 0 ♣
■ A linear combination: (Sk + 1)♥+ Sn♣ = 0(n+ 1) (SnSk − Sk − 1)F = 0
■ Reorganize this to have only Sn and n on the left, and afactor of (Sk − 1) on the right:
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k), for some G(n, k)
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
■ Easy identities: F (n+1,k)F (n,k) = n+1
n−k+1F (n,k+1)
F (n,k) = n−kk+1
■ Cross Multiply:
((n− k + 1)Sn − (n+ 1))F = 0 ♥
((k + 1)Sk − (n− k))F = 0 ♣
■ A linear combination: (Sk + 1)♥+ Sn♣ = 0(n+ 1) (SnSk − Sk − 1)F = 0
■ Reorganize this to have only Sn and n on the left, and afactor of (Sk − 1) on the right:
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k), for some G(n, k)
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
■ Easy identities: F (n+1,k)F (n,k) = n+1
n−k+1F (n,k+1)
F (n,k) = n−kk+1
■ Cross Multiply:
((n− k + 1)Sn − (n+ 1))F = 0 ♥
((k + 1)Sk − (n− k))F = 0 ♣
■ A linear combination: (Sk + 1)♥+ Sn♣ = 0(n+ 1) (SnSk − Sk − 1)F = 0
■ Reorganize this to have only Sn and n on the left, and afactor of (Sk − 1) on the right:
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k), for some G(n, k)
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
■ Easy identities: F (n+1,k)F (n,k) = n+1
n−k+1F (n,k+1)
F (n,k) = n−kk+1
■ Cross Multiply:
((n− k + 1)Sn − (n+ 1))F = 0 ♥
((k + 1)Sk − (n− k))F = 0 ♣
■ A linear combination: (Sk + 1)♥+ Sn♣ = 0(n+ 1) (SnSk − Sk − 1)F = 0
■ Reorganize this to have only Sn and n on the left, and afactor of (Sk − 1) on the right:
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k), for some G(n, k)
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k) (∗)
■ Take the sum:
(n+ 1)(Sn − 2)
a(n) =∑
k≥0
(n+ 1)(Sn − 2)
F (n, k)
=∑
k≥0
(Sk − 1)(G(n, k))
=∑
k≥0
(G(n, k + 1)−G(n, k)) = 0
■ Gives result about the sum:
a(n+ 1)− 2a(n) = 0 =⇒ a(n) = 2n =⇒∑
k
(nk
)= 2n
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k) (∗)
■ Apply the linear combination to this sum:
(n+ 1)(Sn − 2)a(n) =∑
k≥0
(n+ 1)(Sn − 2)F (n, k)
=∑
k≥0
(Sk − 1)(G(n, k))
=∑
k≥0
(G(n, k + 1)−G(n, k)) = 0
■ Gives result about the sum:
a(n+ 1)− 2a(n) = 0 =⇒ a(n) = 2n =⇒∑
k
(nk
)= 2n
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k) (∗)
■ Apply (∗):
(n+ 1)(Sn − 2)a(n) =∑
k≥0
(n+ 1)(Sn − 2)F (n, k)
=∑
k≥0
(Sk − 1)(G(n, k))
=∑
k≥0
(G(n, k + 1)−G(n, k)) = 0
■ Gives result about the sum:
a(n+ 1)− 2a(n) = 0 =⇒ a(n) = 2n =⇒∑
k
(nk
)= 2n
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k) (∗)
■ Expand:
(n+ 1)(Sn − 2)a(n) =∑
k≥0
(n+ 1)(Sn − 2)F (n, k)
=∑
k≥0
(Sk − 1)(G(n, k))
=∑
k≥0
(G(n, k + 1)−G(n, k)) = 0
■ Gives result about the sum:
a(n+ 1)− 2a(n) = 0 =⇒ a(n) = 2n =⇒∑
k
(nk
)= 2n
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 9/26
Zeilberger’s Creative Telescoping
F (n, k) =(nk
)
(n+ 1)(Sn − 2)F = (Sk − 1)G(n, k) (∗)
■ Expand:
(n+ 1)(Sn − 2)a(n) =∑
k≥0
(n+ 1)(Sn − 2)F (n, k)
=∑
k≥0
(Sk − 1)(G(n, k))
=∑
k≥0
(G(n, k + 1)−G(n, k)) = 0
■ Gives result about the sum:
a(n+ 1)− 2a(n) = 0 =⇒ a(n) = 2n =⇒∑
k
(nk
)= 2n
The Set Up
Creative telescoping
● P-Recursive Functions
● Express recurrences as operators
● Elimination
● Zeilberger’s Creative Telescoping
● How does this work in general?
A combinatorial framework
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 10/26
How does this work in general?
■ Begin with a set of “independent” relationsP (n, k, Sn, Sk)F (n, k) = 0 Q(n, k, Sn, Sk)F (n, k) = 0■ Find the good linear combination which eliminates k and has
a factor of Sk − 1
R(n, k, Sn, Sk) = A(n, k, Sn, Sk)P +B(n, k, Sn, Sk)Q
R(n, k, Sn, Sk) = S(Sn, n) + (Sk − 1)R′(n, Sn, Sk)
■ Take the infinite sum over k. We have a telescoping sum:0 =
∑
k R(n, k, Sn, Sk)F (n, k)
0 =∑
k S(Sn, n)F +∑
k(Sk − 1)R′(N,K, n)F︸ ︷︷ ︸
G(n,k)
■ ...And we get a recurrence for the sum
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 11/26
A combinatorial framework
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 12/26
Combinatorial generating functions
Counting information about combinatorial objects can beencoded in formal power series:
Ex. exponential generating function (egf) :
C(z) =∑
n
(number of objects in C of size n)zn
n!
Ex. C = C...
=⇒ C(z) = z + z2/2 + 2z3/3! + 3x4/4! + . . .
Nice: algebraic operations correspond with combinatorialoperations.
■ C = A ∪ B =⇒ C(z) = A(z) +B(z)
■ C = {(a, b), a ∈ A, b ∈ B} =⇒ C(z) = A(z)B(z)
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 12/26
Combinatorial generating functions
Counting information about combinatorial objects can beencoded in formal power series:
Ex. exponential generating function (egf) :
C(z) =∑
n
(number of objects in C of size n)zn
n!
Ex. C = C...
=⇒ C(z) = z + z2/2 + 2z3/3! + 3x4/4! + . . .Nice: algebraic operations correspond with combinatorialoperations.
■ C = A ∪ B =⇒ C(z) = A(z) +B(z)
■ C = {(a, b), a ∈ A, b ∈ B} =⇒ C(z) = A(z)B(z)
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.
xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0,
fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0,
fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite
■∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 13/26
D-finite functions
Def. A function f(x1, x2, . . . , xn) is Differentiably finite (D-finite)with respect to X = x1, . . . , xn if■ For 1 ≤ j ≤ n, f satisfies n linear differential equations with
polynomial coefficients:
φ0(X)f(X) + φ1(X)∂f(X)∂xj
+ . . .+ φk(X)∂kf(X)
∂xkj
= 0
Examples■ f = sin(2x3 + 3yz) is D-finite with respect to {x, y, z}.xfxx − 2fx + 9x5f = 0, fyy − 9z2f = 0, fzz − 9y2f = 0
■ exp(polynomial) is D-finite■
∑f(n)zn D-finite ⇐⇒ f(n) P-recursive
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 14/26
Hierarchy of Power Series
Differentiably Algebraic partitions
D-finite k-regular graphs
Algebraic Dyck Paths
Rationalunrestricted walksin the plane
1−√
1−4x2x
Regular expressionsMotzkin Paths
ex
k × n-latin squares grid walks in the first quadrant
eex−1
Algebraic closure Hadamard product Algebraic substitutionFunctional Inverse Diagonals Hadamard product
Integrals Multiplicative inverse
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 14/26
Hierarchy of Power Series
Differentiably Algebraic partitions
D-finite k-regular graphs
Algebraic Dyck Paths
Rationalunrestricted walksin the plane
1−√
1−4x2x
Regular expressionsMotzkin Paths
ex
k × n-latin squares grid walks in the first quadrant
eex−1
Algebraic closure Hadamard product Algebraic substitutionFunctional Inverse Diagonals Hadamard product
Integrals Multiplicative inverse
The Set Up
Creative telescoping
A combinatorial framework
● Generating functions
● D-finite functions
● D-finiteness
The scalar product
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 14/26
Hierarchy of Power Series
Differentiably Algebraic partitions
D-finite k-regular graphs
Algebraic Dyck Paths
Rationalunrestricted walksin the plane
1−√
1−4x2x
Regular expressionsMotzkin Paths
ex
k × n-latin squares grid walks in the first quadrant
eex−1
Algebraic closure Hadamard product Algebraic substitutionFunctional Inverse Diagonals Hadamard product
Integrals Multiplicative inverse
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 15/26
The scalar product
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 16/26
Symmetric Series
power pk p3 = x31 + x3
2 + x33 + . . .
homogeneous hk h3 = x31 + x3
2 + . . .+ x21x2 + . . .+ x1x2x3 + . . .
elementary en e3 = x1x2x3 + x1x2x4 + x1x3x4 + . . .
monomial mλ m(3,2,1) = x31x
22x3 + x3
2x21x3 + x3
3x22x6 + . . .
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 16/26
Symmetric Series
power pk p3 = x31 + x3
2 + x33 + . . .
homogeneous hk h3 = x31 + x3
2 + . . .+ x21x2 + . . .+ x1x2x3 + . . .
elementary en e3 = x1x2x3 + x1x2x4 + x1x3x4 + . . .
monomial mλ m(3,2,1) = x31x
22x3 + x3
2x21x3 + x3
3x22x6 + . . .
■ pλ = pλ1pλ2
· · · pλk(e.g. p(3,2,2,1) = p3p
22p1)
■ Our algebra of interest: K[[p1, p2, . . . ]]
■ Ex:H =
∑
n hn = exp(∑
npn
n)
E =∑
n en = exp(∑
n(−1)n+1 pn
n)
■ Scalar product: 〈pλ, pµ〉 = δλµzλ
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 16/26
Symmetric Series
power pk p3 = x31 + x3
2 + x33 + . . .
homogeneous hk h3 = x31 + x3
2 + . . .+ x21x2 + . . .+ x1x2x3 + . . .
elementary en e3 = x1x2x3 + x1x2x4 + x1x3x4 + . . .
monomial mλ m(3,2,1) = x31x
22x3 + x3
2x21x3 + x3
3x22x6 + . . .
■ pλ = pλ1pλ2
· · · pλk(e.g. p(3,2,2,1) = p3p
22p1)
■ Our algebra of interest: K[[p1, p2, . . . ]]
■ Ex:H =
∑
n hn = exp(∑
npn
n)
E =∑
n en = exp(∑
n(−1)n+1 pn
n)
■ Scalar product: 〈pλ, pµ〉 = δλµzλ
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 16/26
Symmetric Series
power pk p3 = x31 + x3
2 + x33 + . . .
homogeneous hk h3 = x31 + x3
2 + . . .+ x21x2 + . . .+ x1x2x3 + . . .
elementary en e3 = x1x2x3 + x1x2x4 + x1x3x4 + . . .
monomial mλ m(3,2,1) = x31x
22x3 + x3
2x21x3 + x3
3x22x6 + . . .
■ pλ = pλ1pλ2
· · · pλk(e.g. p(3,2,2,1) = p3p
22p1)
■ Our algebra of interest: K[[p1, p2, . . . ]]
■ Ex:H =
∑
n hn = exp(∑
npn
n)
E =∑
n en = exp(∑
n(−1)n+1 pn
n)
■ Scalar product: 〈pλ, pµ〉 = δλµzλ
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 16/26
Symmetric Series
power pk p3 = x31 + x3
2 + x33 + . . .
homogeneous hk h3 = x31 + x3
2 + . . .+ x21x2 + . . .+ x1x2x3 + . . .
elementary en e3 = x1x2x3 + x1x2x4 + x1x3x4 + . . .
monomial mλ m(3,2,1) = x31x
22x3 + x3
2x21x3 + x3
3x22x6 + . . .
■ pλ = pλ1pλ2
· · · pλk(e.g. p(3,2,2,1) = p3p
22p1)
■ Our algebra of interest: K[[p1, p2, . . . ]]
■ Ex:H =
∑
n hn = exp(∑
npn
n)
E =∑
n en = exp(∑
n(−1)n+1 pn
n)
■ Scalar product: 〈pλ, pµ〉 = δλµzλ
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 17/26
D-finite symmetric series
Def. A symmetric series F ∈ K[[p1, p2, . . . ; t]] is D-finite if forany n,■ F (p1, ..., pn, 0, 0, . . . ; t) is D-finite wrt p1, . . . , pn, t.
■ Examples:
H =∑hnt
n = exp(∑
n pntn/n)
E =∑ent
n = exp(∑
n(−1)n+1pntn/n)
S =∑
λ sλt|λ| = exp(
∑
n
p2
nt2n
2n+ p2n−1t2n−1
2n−1 )
∑
λ all parts odd sλt|λ| = SE−1
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 17/26
D-finite symmetric series
Def. A symmetric series F ∈ K[[p1, p2, . . . ; t]] is D-finite if forany n,■ F (p1, ..., pn, 0, 0, . . . ; t) is D-finite wrt p1, . . . , pn, t.■ Examples:
H =∑hnt
n = exp(∑
n pntn/n)
E =∑ent
n = exp(∑
n(−1)n+1pntn/n)
S =∑
λ sλt|λ| = exp(
∑
n
p2
nt2n
2n+ p2n−1t2n−1
2n−1 )
∑
λ all parts odd sλt|λ| = SE−1
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 18/26
The Scalar Product
■ The scalar product, (symmetric, bilinear form) is defined by
〈pλ, pµ〉 = δλµzλ.
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
■ Equivalent to coefficient extraction:
[xk1
1 xk2
2 . . . xkn
n ]F = 〈F, hk1hk2
· · ·hkn〉
■ Adjunction:〈pnF,G〉 = 〈F, n ∂
∂pnG〉.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 18/26
The Scalar Product
■ The scalar product, (symmetric, bilinear form) is defined by
〈pλ, pµ〉 = δλµzλ.
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
■ Equivalent to coefficient extraction:
[xk1
1 xk2
2 . . . xkn
n ]F = 〈F, hk1hk2
· · ·hkn〉
■ Adjunction:〈pnF,G〉 = 〈F, n ∂
∂pnG〉.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 18/26
The Scalar Product
■ The scalar product, (symmetric, bilinear form) is defined by
〈pλ, pµ〉 = δλµzλ.
λ = (1m12m2 · · · kmk ) =⇒ zλ = 1m1m1!2m2m2! · · · kmkmk!
■ Equivalent to coefficient extraction:
[xk1
1 xk2
2 . . . xkn
n ]F = 〈F, hk1hk2
· · ·hkn〉
■ Adjunction:〈pnF,G〉 = 〈F, n ∂
∂pnG〉.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 19/26
The Scalar Product
Theorem (Gessel, 90): If F ∈ K[[p1, p2, . . . , pn; t]] and G ∈ K[[p1, p2, . . . ; t]]are both D-finite symmetric series then the scalar product
〈F,G〉 = φ(t)
is a D-finite function with respect to t.
That is, φ(t) satisfies a linear DE with polynomial coefficients.Infact, this DE can be calculated (Chyzak, M., Salvy 02)
Termination guaranteed by D-finiteness.
Ex:> scalar de(exp(p1+p12/2+p2+p22),exp((p22+p2/2)*t),[t],f);
[(−126 t+ 13− 16 t2
)f (t) +
(−1 + 32 t− 256 t2
)ddtf (t)]
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 19/26
The Scalar Product
Theorem (Gessel, 90): If F ∈ K[[p1, p2, . . . , pn; t]] and G ∈ K[[p1, p2, . . . ; t]]are both D-finite symmetric series then the scalar product
〈F,G〉 = φ(t)
is a D-finite function with respect to t.
That is, φ(t) satisfies a linear DE with polynomial coefficients.Infact, this DE can be calculated (Chyzak, M., Salvy 02)
Termination guaranteed by D-finiteness.
Ex:> scalar de(exp(p1+p12/2+p2+p22),exp((p22+p2/2)*t),[t],f);
[(−126 t+ 13− 16 t2
)f (t) +
(−1 + 32 t− 256 t2
)ddtf (t)]
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 20/26
Encoding combinatorial objects
Object Encoding π Sum over objects in the class
MAKE SPACE
t
,,
eee
ll
%%%
,,
t t
tt
t
t2 3
4
6 7
5
1
x31x
42x
23x
54x
55x
26x
27
G =∑
g a graph π(g)
= exp(∑
n
p2
n
2n− p2n
2n)
Set(C) x21x
22x
23x4x5 H[ZC], E[ZC]
C a labelled,finite comb. class
xk
i=⇒ k occurrences
of label i
H =
�
hn, E =
�
en,Z = Polya cycle index,[]= plethystic substitution
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 20/26
Encoding combinatorial objects
Object Encoding π Generating function of regular subclass
MAKE SPACE
t
t
t
t2
4 3
1
x21x
22x
23x
24 Gk(t) = 〈G,
∑hn
2 tn〉
■ Idea: Encode all objects, use scalar product to extract the number with acertain regularity
■ Scalar product with a series gives the egf of a “regular” subclass.
■ Other examples: Young tableaux, k × n-latin squares, hypergraphs
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 21/26
Weyl Algebras
Ap := K〈p1, . . . , pn, ∂1, . . . , ∂n〉
∂ipi = pi∂i + 1, all other variables commute
f(p1, . . . , pn)
Left annihilating ideal: If
= {φ ∈ Ap|φ · f = 0}
Left module: Apf
= {ψ · f |ψ ∈ Ap} ' Ap/If
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 21/26
Weyl Algebras
Ap := K〈p1, . . . , pn, ∂1, . . . , ∂n〉
∂ipi = pi∂i + 1, all other variables commute
f(p1, . . . , pn)
Left annihilating ideal: If
= {φ ∈ Ap|φ · f = 0}
Left module: Apf
= {ψ · f |ψ ∈ Ap} ' Ap/If
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 21/26
Weyl Algebras
Ap := K〈p1, . . . , pn, ∂1, . . . , ∂n〉
∂ipi = pi∂i + 1, all other variables commute
f(p1, . . . , pn) f = sin(p1 + p2/2)
Left annihilating ideal: If
= {φ ∈ Ap|φ · f = 0} If = Ap〈∂21 + 1, 4∂2
2 + 1, 2∂1 − ∂2〉
Left module: Apf
= {ψ · f |ψ ∈ Ap} ' Ap/If Apf = K[p1, p2]sin(p1 + p2/2)
⊕K[p1, p2]cos(p1 + p2/2)
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG〈F, φ⊥ ·G〉 − 〈F, ψ ·G〉 = 0 p⊥n = n∂n, ∂
⊥n = pn/n
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG〈F, φ⊥ ·G〉 − 〈F, ψ ·G〉 = 0 p⊥n = n∂n, ∂
⊥n = pn/n
〈F,(φ⊥ − ψ
)·G〉 = 0 by linearity
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG〈F, φ⊥ ·G〉 − 〈F, ψ ·G〉 = 0 p⊥n = n∂n, ∂
⊥n = pn/n
〈F,(φ⊥ − ψ
)·G〉 = 0 by linearity
If,(φ⊥ − ψ
)= γ(t, ∂t) ∈ C[t, ∂t], then...
〈F, γ(t, ∂t) ·G〉 = 0
=⇒ γ(t, ∂t) · 〈F,G〉 = 0
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG〈F, φ⊥ ·G〉 − 〈F, ψ ·G〉 = 0 p⊥n = n∂n, ∂
⊥n = pn/n
〈F,(φ⊥ − ψ
)·G〉 = 0 by linearity
If,(φ⊥ − ψ
)= γ(t, ∂t) ∈ C[t, ∂t], then...
〈F, γ(t, ∂t) ·G〉 = 0 =⇒ γ(t, ∂t) · 〈F,G〉 = 0
This gives a differential equation satisfied by 〈F,G〉.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 22/26
How does it work?
Like with creative telescoping, we take linear combinations ofequations to eliminate variables and generate new equations.0− 0 = 0〈0, G〉 − 〈F, 0〉 = 0〈φ · F,G〉 − 〈F, ψ ·G〉 = 0 φ ∈ IF , ψ ∈ IG〈F, φ⊥ ·G〉 − 〈F, ψ ·G〉 = 0 p⊥n = n∂n, ∂
⊥n = pn/n
〈F,(φ⊥ − ψ
)·G〉 = 0 by linearity
If,(φ⊥ − ψ
)= γ(t, ∂t) ∈ C[t, ∂t], then...
〈F, γ(t, ∂t) ·G〉 = 0 =⇒ γ(t, ∂t) · 〈F,G〉 = 0
This gives a differential equation satisfied by 〈F,G〉.
Any non-trivial element γ of(
I⊥f + Ig
)⋂C[t, ∂t] provides a
differential equation satisfied by 〈f, g〉.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 23/26
Holonomic Modules
Ax := K〈x1, . . . , xn, ∂1, . . . , ∂n〉∂ixi = xi∂i + 1, all other variables commute
Filtration: A(d)x = {ψ ∈ Ax| degψ ≤ d}
Def. A module Axf is holonomic if
dimA(d)x · f = O(dn).
✔ K[x1, ..., xn] = Ax • 1: dimA(d)x · 1 = O(dn)
✘ Ax: dimA(d)x = O(d2n)
✔ Ax · sin(x) ' K[x] sin(x)⊕K[x] cos(x)
dimA(d)x · sin(x) = 2 dimK[x](d) = O(d1)
Theorem: (Kashiwara) f(x) D-finite ⇐⇒ Axf holonomic
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 23/26
Holonomic Modules
Ax := K〈x1, . . . , xn, ∂1, . . . , ∂n〉∂ixi = xi∂i + 1, all other variables commute
Filtration: A(d)x = {ψ ∈ Ax| degψ ≤ d}
Def. A module Axf is holonomic if
dimA(d)x · f = O(dn).
✔ K[x1, ..., xn] = Ax • 1: dimA(d)x · 1 = O(dn)
✘ Ax: dimA(d)x = O(d2n)
✔ Ax · sin(x) ' K[x] sin(x)⊕K[x] cos(x)
dimA(d)x · sin(x) = 2 dimK[x](d) = O(d1)
Theorem: (Kashiwara) f(x) D-finite ⇐⇒ Axf holonomic
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 23/26
Holonomic Modules
Ax := K〈x1, . . . , xn, ∂1, . . . , ∂n〉∂ixi = xi∂i + 1, all other variables commute
Filtration: A(d)x = {ψ ∈ Ax| degψ ≤ d}
Def. A module Axf is holonomic if
dimA(d)x · f = O(dn).
✔ K[x1, ..., xn] = Ax • 1: dimA(d)x · 1 = O(dn)
✘ Ax: dimA(d)x = O(d2n)
✔ Ax · sin(x) ' K[x] sin(x)⊕K[x] cos(x)
dimA(d)x · sin(x) = 2 dimK[x](d) = O(d1)
Theorem: (Kashiwara) f(x) D-finite ⇐⇒ Axf holonomic
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 23/26
Holonomic Modules
Ax := K〈x1, . . . , xn, ∂1, . . . , ∂n〉∂ixi = xi∂i + 1, all other variables commute
Filtration: A(d)x = {ψ ∈ Ax| degψ ≤ d}
Def. A module Axf is holonomic if
dimA(d)x · f = O(dn).
✔ K[x1, ..., xn] = Ax • 1: dimA(d)x · 1 = O(dn)
✘ Ax: dimA(d)x = O(d2n)
✔ Ax · sin(x) ' K[x] sin(x)⊕K[x] cos(x)
dimA(d)x · sin(x) = 2 dimK[x](d) = O(d1)
Theorem: (Kashiwara) f(x) D-finite ⇐⇒ Axf holonomic
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 23/26
Holonomic Modules
Ax := K〈x1, . . . , xn, ∂1, . . . , ∂n〉∂ixi = xi∂i + 1, all other variables commute
Filtration: A(d)x = {ψ ∈ Ax| degψ ≤ d}
Def. A module Axf is holonomic if
dimA(d)x · f = O(dn).
✔ K[x1, ..., xn] = Ax • 1: dimA(d)x · 1 = O(dn)
✘ Ax: dimA(d)x = O(d2n)
✔ Ax · sin(x) ' K[x] sin(x)⊕K[x] cos(x)
dimA(d)x · sin(x) = 2 dimK[x](d) = O(d1)
Theorem: (Kashiwara) f(x) D-finite ⇐⇒ Axf holonomic
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 24/26
(I⊥f + Ig
)∩ At is not empty!
Idea: model adjunction p⊥n = n∂pnby a tensor product
■ Define surjection:Ap,tf ⊗
⊥C[t] Ap,tg −→ Ap,t〈f, g〉
a⊗⊥C[t] b 7→ 〈a, b〉
Prop: I⊥f + Ig contains AnnAt(f ⊗⊥C[t] g)
■ Thm: Ap,tf ⊗⊥C[t] Ap,tg is a holonomic Ap,t-module
■
(
Ap,tf ⊗⊥C[t] Ap,tg
)
∩ “At” is a holonomic At-module
isomorphic to At/AnnAt(f ⊗⊥
C[t] g) as At-modules
Cor: AnnAt(f ⊗⊥
C[t] g) is non-trivial.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 24/26
(I⊥f + Ig
)∩ At is not empty!
Idea: model adjunction p⊥n = n∂pnby a tensor product
■ Define surjection:Ap,tf ⊗
⊥C[t] Ap,tg −→ Ap,t〈f, g〉
a⊗⊥C[t] b 7→ 〈a, b〉
Prop: I⊥f + Ig contains AnnAt(f ⊗⊥C[t] g)
■ Thm: Ap,tf ⊗⊥C[t] Ap,tg is a holonomic Ap,t-module
key fact: ⊥ is degree preserving
■
(
Ap,tf ⊗⊥C[t] Ap,tg
)
∩ “At” is a holonomic At-module
isomorphic to At/AnnAt(f ⊗⊥
C[t] g) as At-modules
Cor: AnnAt(f ⊗⊥
C[t] g) is non-trivial.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 24/26
(I⊥f + Ig
)∩ At is not empty!
Idea: model adjunction p⊥n = n∂pnby a tensor product
■ Define surjection:Ap,tf ⊗
⊥C[t] Ap,tg −→ Ap,t〈f, g〉
a⊗⊥C[t] b 7→ 〈a, b〉
Prop: I⊥f + Ig contains AnnAt(f ⊗⊥C[t] g)
■ Thm: Ap,tf ⊗⊥C[t] Ap,tg is a holonomic Ap,t-module
■
(
Ap,tf ⊗⊥C[t] Ap,tg
)
∩ “At” is a holonomic At-module
isomorphic to At/AnnAt(f ⊗⊥
C[t] g) as At-modules
Cor: AnnAt(f ⊗⊥
C[t] g) is non-trivial.
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
● Symmetric Series
● D-finite symmetric series
● The Scalar Product
● The Scalar Product
● Encoding combinatorial objects
● Weyl Algebras
● How does it work?
● Holonomic Modules
● Proof
● Other applications
Conclusion
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 25/26
Other applications
Algorithm computes scalar products whose adjoint operation isa degree preserving operation:➢Related to Macdonald Polynomials
〈pλ, pµ〉q,t = zλδµλ
∏
λi∈λ
1− qλi
1− tλi
➢MacMahon symmetric functions
〈p(a,b)2(c,d), p(a,b)2(c,d)〉 = 2
(a!b!
(a+ b− 1)!
)2 (c!d!
(c+ d− 1)!
)
The Set Up
Creative telescoping
A combinatorial framework
The scalar product
Conclusion
● Lots to do...
Marni Mishna, Freaky Friday the 13th, 2004 Carleton University Holonomy and Combinatorics – p. 26/26
Lots to do...
■ Other D-finite closure properties?■ Effective closure properties??■ Classifying families of combinatorial objects e.g. walks in the
plane, families of generalized partitions■ Improving the Gröbner basis calculations