Binomial Determinants for Tiling Problems Yield to the Holonomic Ansatz
Hao Du, Christoph Koutschan, Thotsaporn Thanatipanonda, Elaine Wong
Johann Radon Institute for Computational and Applied Mathematics (RICAM) Austrian Academy of Sciences
July 2, 2021 S´eminairede Combinatoire de Lyon `al’ENS + 1 + 1 + 1
µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
Example: is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 µ+9 µ+10 µ+11 µ+12 µ+13 µ+10 µ+11 µ+12 µ+13 µ+14 µ+11 µ+12 µ+13 µ+14 µ+15 µ+12 µ+13 µ+14 µ+15 µ+16
Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n
1 / 32 + 1 + 1 + 1
Example: is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 µ+9 µ+10 µ+11 µ+12 µ+13 µ+10 µ+11 µ+12 µ+13 µ+14 µ+11 µ+12 µ+13 µ+14 µ+15 µ+12 µ+13 µ+14 µ+15 µ+16
Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
1 / 32 + 1 + 1 + 1
Example: is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 µ+9 µ+10 µ+11 µ+12 µ+13 µ+10 µ+11 µ+12 µ+13 µ+14 µ+11 µ+12 µ+13 µ+14 µ+15 µ+12 µ+13 µ+14 µ+15 µ+16
Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ History: D0,0(n) was introduced in the work of Andrews in 1979 – 1980 in the context of descending plane partitions:
1 / 32 Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ Example: D4,6(5) is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 6 7 8 9 10 µ+9 µ+10 µ+11 µ+12 µ+13 6 7 8 9 10 µ+10 + 1 µ+11 µ+12 µ+13 µ+14 6 7 8 9 10 µ+11 µ+12 µ+13 µ+14 µ+15 + 1 6 7 8 9 10 µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 + 1 9 10 1 / 32 Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ+2 Example: D3,5 (5) is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 5 6 7 8 9 µ+9 µ+10 µ+11 µ+12 µ+13 5 6 7 8 9 µ+10 + 1 µ+11 µ+12 µ+13 µ+14 5 6 7 8 9 µ+11 µ+12 µ+13 µ+14 µ+15 + 1 5 6 7 8 9 µ+12 µ+13 µ+14 µ+15 µ+16 5 6 7 + 1 8 9 1 / 32 Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ Example: D4,6(5) is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 6 7 8 9 10 µ+9 µ+10 µ+11 µ+12 µ+13 6 7 8 9 10 µ+10 + 1 µ+11 µ+12 µ+13 µ+14 6 7 8 9 10 µ+11 µ+12 µ+13 µ+14 µ+15 + 1 6 7 8 9 10 µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 + 1 9 10 1 / 32 + 1 + 1 + 1
Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ−1 Example: D5,6 (5) is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 6 7 8 9 10 µ+9 + 1 µ+10 µ+11 µ+12 µ+13 6 7 8 9 10 µ+10 µ+11 + 1 µ+12 µ+13 µ+14 6 7 8 9 10 µ+11 µ+12 µ+13 µ+14 µ+15 + 1 6 7 8 9 10 µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 9 + 1 10 1 / 32 + 1 + 1 + 1
Families of Binomial Determinants Definition: For n ∈ N, for s, t ∈ Z, and for µ an indeterminate, define the following (n × n)-determinants: µ µ + i + j + s + t − 4 Ds,t(n) := det + δi+s,j+t , 16i6n j + t − 1 16j6n µ µ + i + j + s + t − 4 Es,t(n) := det − δi+s,j+t . 16i6n j + t − 1 16j6n
µ−1 Example: E5,6 (5) is the determinant of the matrix µ+8 µ+9 µ+10 µ+11 µ+12 6 7 8 9 10 µ+9 − 1 µ+10 µ+11 µ+12 µ+13 6 7 8 9 10 µ+10 µ+11 − 1 µ+12 µ+13 µ+14 6 7 8 9 10 µ+11 µ+12 µ+13 µ+14 µ+15 − 1 6 7 8 9 10 µ+12 µ+13 µ+14 µ+15 µ+16 6 7 8 9 − 1 10 1 / 32 “Conjecture 37” (Lascoux/Krattenthaler 2005) Let µ be an indeterminate and m, r ∈ Z. If m > r > 1, then
µ m−r 4m+(m−r)(m−r−1)−3r E1,2r−1(2m − 1) = (−1) · 2 2r−3 ! m−1 ! Y Y i!(i + 1)! × i! · (2i)! (2i + 2)! i=0 i=0 2r−2 ! µ 1 Y × (µ − 1) · 2 + r − 2 m−r · (µ + i − 1)2m+2r−2i−1 i=1 r−2 2 ! Y (2m − 2i − 3)! × 2 i=0 (m − i − 2)! (2m + 2i − 1)! (2m + 2i + 1)! m−r−1 b 2 c Y 2 2 × µ + 3i + 3r − 1 − µ − 3m + 3i + 3 2 2 m−r−2i−1 2 m−r−2i i=0
2 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
s s
t t
n even n odd
s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
s s
t t
n even n odd
s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
s s
t t
n even n odd
s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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t t
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s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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t t
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s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
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s s
3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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3 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
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3 / 32 For each path P , let ω(P ) be the product of its edge weights. Let X e(a, b) = ω(P ) and P :a→b e(a1, b1) e(a1, b2) ··· e(a1, bn) e(a2, b1) e(a2, b2) ··· e(a2, bn) M = . . . .. . . . . . e(an, b1) e(an, b2) ··· e(an, bn) Then the determinant of M is the signed sum over all n-tuples P = (P1,...,Pn) of non-intersecting paths from A to B: n X Y det(M) = sign(σ(P )) ω(Pi).
(P1,...,Pn): A→B i=1 where σ denotes a permutation that is applied to B.
Lindstr¨om-Gessel-ViennotLemma Let G be a directed acyclic graph and consider base vertices A = {a1, . . . , an} and destination vertices B = {b1, . . . , bn}.
4 / 32 Then the determinant of M is the signed sum over all n-tuples P = (P1,...,Pn) of non-intersecting paths from A to B: n X Y det(M) = sign(σ(P )) ω(Pi).
(P1,...,Pn): A→B i=1 where σ denotes a permutation that is applied to B.
Lindstr¨om-Gessel-ViennotLemma Let G be a directed acyclic graph and consider base vertices A = {a1, . . . , an} and destination vertices B = {b1, . . . , bn}. For each path P , let ω(P ) be the product of its edge weights. Let X e(a, b) = ω(P ) and P :a→b e(a1, b1) e(a1, b2) ··· e(a1, bn) e(a2, b1) e(a2, b2) ··· e(a2, bn) M = . . . .. . . . . . e(an, b1) e(an, b2) ··· e(an, bn)
4 / 32 Lindstr¨om-Gessel-ViennotLemma Let G be a directed acyclic graph and consider base vertices A = {a1, . . . , an} and destination vertices B = {b1, . . . , bn}. For each path P , let ω(P ) be the product of its edge weights. Let X e(a, b) = ω(P ) and P :a→b e(a1, b1) e(a1, b2) ··· e(a1, bn) e(a2, b1) e(a2, b2) ··· e(a2, bn) M = . . . .. . . . . . e(an, b1) e(an, b2) ··· e(an, bn) Then the determinant of M is the signed sum over all n-tuples P = (P1,...,Pn) of non-intersecting paths from A to B: n X Y det(M) = sign(σ(P )) ω(Pi).
(P1,...,Pn): A→B i=1 where σ denotes a permutation that is applied to B. 4 / 32 t + n − 1
. .
t + 1
t
µ + s − 2 ... µ + s + n − 3
Lindstr¨om-Gessel-ViennotLemma In our context, it implies that the determinant without the Kronecker delta µ + i + j + s + t − 4 det 16i,j6n j + t − 1 counts n-tuples of non-intersecting paths in the lattice N2:
5 / 32 Lindstr¨om-Gessel-ViennotLemma In our context, it implies that the determinant without the Kronecker delta µ + i + j + s + t − 4 det 16i,j6n j + t − 1 counts n-tuples of non-intersecting paths in the lattice N2:
t + n − 1
. .
t + 1
t
µ + s − 2 ... µ + s + n − 3
5 / 32 Lindstr¨om-Gessel-ViennotLemma In our context, it implies that the determinant without the Kronecker delta µ + i + j + s + t − 4 det 16i,j6n j + t − 1 counts n-tuples of non-intersecting paths in the lattice N2:
t + n − 1
. .
t + 1
t
µ + s − 2 ... µ + s + n − 3
5 / 32 Lattice Paths ←→ Rhombus Tilings
6 / 32 Lattice Paths ←→ Rhombus Tilings
6 / 32 Lattice Paths ←→ Rhombus Tilings
6 / 32 Lattice Paths ←→ Rhombus Tilings
6 / 32 By applying this procedure recursively, one obtains
X I = det BI+s−t (s > t), I⊆{1,...,n−s+t}
I where BJ denotes the matrix that is obtained by deleting all rows with indices in I and all columns with indices in J from the matrix µ + i + j + s + t − 4 B = j + t − 1 16i,j6n
Expansion of Kronecker Deltas
··· b1,6 b1,7 + 1 b1,8 ···
··· b b b + 1 ··· Laplace expansion: 2,6 2,7 2,8 = ......
··· b1,6 b1,7 b1,8 ··· ··· b2,6 b2,8 + 1 ··· ··· b b b + 1 ··· 2,6 2,7 2,8 ± ......
7 / 32 I where BJ denotes the matrix that is obtained by deleting all rows with indices in I and all columns with indices in J from the matrix µ + i + j + s + t − 4 B = j + t − 1 16i,j6n
Expansion of Kronecker Deltas
··· b1,6 b1,7 + 1 b1,8 ···
··· b b b + 1 ··· Laplace expansion: 2,6 2,7 2,8 = ......
··· b1,6 b1,7 b1,8 ··· ··· b2,6 b2,8 + 1 ··· ··· b b b + 1 ··· 2,6 2,7 2,8 ± ...... By applying this procedure recursively, one obtains
µ X (s−t)·|I| I Ds,t(n) = (−1) det BI+s−t (s > t), I⊆{1,...,n−s+t}
7 / 32 Expansion of Kronecker Deltas
··· b1,6 b1,7 + 1 b1,8 ···
··· b b b + 1 ··· Laplace expansion: 2,6 2,7 2,8 = ......
··· b1,6 b1,7 b1,8 ··· ··· b2,6 b2,8 + 1 ··· ··· b b b + 1 ··· 2,6 2,7 2,8 ± ...... By applying this procedure recursively, one obtains
µ X (s−t)·|I| I Ds,t(n) = (−1) det BI+s−t (s > t), I⊆{1,...,n−s+t}
I where BJ denotes the matrix that is obtained by deleting all rows with indices in I and all columns with indices in J from the matrix µ + i + j + s + t − 4 B = j + t − 1 16i,j6n 7 / 32 Expansion of Kronecker Deltas
··· b1,6 b1,7 + 1 b1,8 ···
··· b b b + 1 ··· Laplace expansion: 2,6 2,7 2,8 = ......
··· b1,6 b1,7 b1,8 ··· ··· b2,6 b2,8 + 1 ··· ··· b b b + 1 ··· 2,6 2,7 2,8 ± ...... By applying this procedure recursively, one obtains
µ X (s−t+1)·|I| I Es,t(n) = (−1) det BI+s−t (s > t), I⊆{1,...,n−s+t}
I where BJ denotes the matrix that is obtained by deleting all rows with indices in I and all columns with indices in J from the matrix µ + i + j + s + t − 4 B = j + t − 1 16i,j6n 7 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Hexagonal Fusion s = 1 t = 3 n = 6 µ = 5
8 / 32 Combinatorial Interpretation: Holey Hexagon
s = 5 t = 7 n = 8 µ = 8
9 / 32 Combinatorial Interpretation: Holey Hexagon
s = 5 t = 7 n = 8 µ = 8
9 / 32 Combinatorial Interpretation: Holey Hexagon
s = 5 t = 7 n = 8 µ = 8
9 / 32 Proof (by example): 3 6 D2,0(4) E1,0(3)
A Combinatorial Proof Lemma: For n, s ∈ Z such that n > s > 1 and n > 1, µ µ+3 Es,0(n) = Ds−1,0(n − 1), µ µ+3 Ds,0(n) = Es−1,0(n − 1).
10 / 32 A Combinatorial Proof Lemma: For n, s ∈ Z such that n > s > 1 and n > 1, µ µ+3 Es,0(n) = Ds−1,0(n − 1), µ µ+3 Ds,0(n) = Es−1,0(n − 1).
Proof (by example): 3 6 D2,0(4) E1,0(3)
10 / 32 A Combinatorial Proof Lemma: For n, s ∈ Z such that n > s > 1 and n > 1, µ µ+3 Es,0(n) = Ds−1,0(n − 1), µ µ+3 Ds,0(n) = Es−1,0(n − 1).
Proof (by example): 3 6 D2,0(4) E1,0(3)
10 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t
n odd n even
s s
t t
n even n odd
s s
11 / 32 Theorem (Desnanot-Jacobi-Dodgson). Let mi,j i,j∈Z and Ms,t(n) := dets6i Ms,t(n)Ms+1,t+1(n − 2) = Ms,t(n − 1)Ms+1,t+1(n − 1) − Ms+1,t(n − 1)Ms,t+1(n − 1). Schematically: × = × − × Switching Lemma and DJD µ µ µ Lemma: Let As,t(n) be either Ds,t(n) or Es,t(n). For real numbers s, t∈ / {−1, −2,...} with t − s ∈ N and n ∈ Z+, t−s−1 Y µ + s + i − 1 Aµ (n) = n · Aµ (n). s,t i + s + 1 t,s i=0 n 12 / 32 Schematically: × = × − × Switching Lemma and DJD µ µ µ Lemma: Let As,t(n) be either Ds,t(n) or Es,t(n). For real numbers s, t∈ / {−1, −2,...} with t − s ∈ N and n ∈ Z+, t−s−1 Y µ + s + i − 1 Aµ (n) = n · Aµ (n). s,t i + s + 1 t,s i=0 n Theorem (Desnanot-Jacobi-Dodgson). Let mi,j i,j∈Z and Ms,t(n) := dets6i Ms,t(n)Ms+1,t+1(n − 2) = Ms,t(n − 1)Ms+1,t+1(n − 1) − Ms+1,t(n − 1)Ms,t+1(n − 1). 12 / 32 Switching Lemma and DJD µ µ µ Lemma: Let As,t(n) be either Ds,t(n) or Es,t(n). For real numbers s, t∈ / {−1, −2,...} with t − s ∈ N and n ∈ Z+, t−s−1 Y µ + s + i − 1 Aµ (n) = n · Aµ (n). s,t i + s + 1 t,s i=0 n Theorem (Desnanot-Jacobi-Dodgson). Let mi,j i,j∈Z and Ms,t(n) := dets6i Ms,t(n)Ms+1,t+1(n − 2) = Ms,t(n − 1)Ms+1,t+1(n − 1) − Ms+1,t(n − 1)Ms,t+1(n − 1). Schematically: × = × − × 12 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 13 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 13 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 13 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 13 / 32 Theorem: For m, r ∈ Z and m > r > 1, m−r µ (−1) (µ − 1) µ + 2r − 1 2m−2 E2r−1,1(2m − 1) = µ (2r − 2)! m + r − 1 m−r+1 2 + r m−r m−r 2 µ 2 Y µ + 2i + 6r − 5 2 + 2i + 3r − 2 × i−1 i . 2 µ 2 i=1 i i 2 + i + 3r − 2 i−1 µ Corollary: Apply switching lemma to obtain E1,2r−1(2m − 1). Proof of the Lascoux/Krattenthaler Conjecture Lemma: For m, r ∈ Z and m > r > 1, µ D (2m) (m + r − 1)(µ − 1)(µ + 2m + 1)(µ + 2r) 2r,1 = , µ+3 2m(2r − 1)(µ + 2)(µ + 2m + 2r − 1) E2r−1,1(2m − 1) µ E (2m + 1) (m + r)(µ − 1)(µ + 2m + 2)(µ + 2r + 1) 2r+1,1 = . µ+3 2r(2m + 1)(µ + 2)(µ + 2m + 2r + 1) D2r,1 (2m) 14 / 32 µ Corollary: Apply switching lemma to obtain E1,2r−1(2m − 1). Proof of the Lascoux/Krattenthaler Conjecture Lemma: For m, r ∈ Z and m > r > 1, µ D (2m) (m + r − 1)(µ − 1)(µ + 2m + 1)(µ + 2r) 2r,1 = , µ+3 2m(2r − 1)(µ + 2)(µ + 2m + 2r − 1) E2r−1,1(2m − 1) µ E (2m + 1) (m + r)(µ − 1)(µ + 2m + 2)(µ + 2r + 1) 2r+1,1 = . µ+3 2r(2m + 1)(µ + 2)(µ + 2m + 2r + 1) D2r,1 (2m) Theorem: For m, r ∈ Z and m > r > 1, m−r µ (−1) (µ − 1) µ + 2r − 1 2m−2 E2r−1,1(2m − 1) = µ (2r − 2)! m + r − 1 m−r+1 2 + r m−r m−r 2 µ 2 Y µ + 2i + 6r − 5 2 + 2i + 3r − 2 × i−1 i . 2 µ 2 i=1 i i 2 + i + 3r − 2 i−1 14 / 32 Proof of the Lascoux/Krattenthaler Conjecture Lemma: For m, r ∈ Z and m > r > 1, µ D (2m) (m + r − 1)(µ − 1)(µ + 2m + 1)(µ + 2r) 2r,1 = , µ+3 2m(2r − 1)(µ + 2)(µ + 2m + 2r − 1) E2r−1,1(2m − 1) µ E (2m + 1) (m + r)(µ − 1)(µ + 2m + 2)(µ + 2r + 1) 2r+1,1 = . µ+3 2r(2m + 1)(µ + 2)(µ + 2m + 2r + 1) D2r,1 (2m) Theorem: For m, r ∈ Z and m > r > 1, m−r µ (−1) (µ − 1) µ + 2r − 1 2m−2 E2r−1,1(2m − 1) = µ (2r − 2)! m + r − 1 m−r+1 2 + r m−r m−r 2 µ 2 Y µ + 2i + 6r − 5 2 + 2i + 3r − 2 × i−1 i . 2 µ 2 i=1 i i 2 + i + 3r − 2 i−1 µ Corollary: Apply switching lemma to obtain E1,2r−1(2m − 1). 14 / 32 Proof of the Lascoux/Krattenthaler Conjecture Lemma: For n, s ∈ Z and n > s > 1, µ A (n) (n + s − 2)(µ − 1)(µ + n + 1)(µ + s) s,1 = , µ+3 2n(s − 1)(µ + 2)(µ + n + s − 1) Bs−1,1(n − 1) where (A, B, s, n) is (D,E, 2r, 2m) or (E,D, 2r + 1, 2m + 1). Theorem: For m, r ∈ Z and m > r > 1, m−r µ (−1) (µ − 1) µ + 2r − 1 2m−2 E2r−1,1(2m − 1) = µ (2r − 2)! m + r − 1 m−r+1 2 + r m−r m−r 2 µ 2 Y µ + 2i + 6r − 5 2 + 2i + 3r − 2 × i−1 i . 2 µ 2 i=1 i i 2 + i + 3r − 2 i−1 µ Corollary: Apply switching lemma to obtain E1,2r−1(2m − 1). 14 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 0 0 0 0 1 0 0 µ 0 1 0 0 ·D (4) · = 0 0 1 0 2,1 0 0 1 0 0 0 0 1 0 0 0 1 µ+1 µ+2 µ+3 µ+4 1 2 + 1 3 4 µ+2 µ+3 µ+4 + 1 µ+5 1 2 3 4 µ+3 µ+4 µ+5 µ+6 + 1 1 2 3 4 µ+4 µ+5 µ+6 µ+7 1 2 3 4 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 0 0 0 −1 1 0 0 µ 0 1 0 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+2 µ+3 µ+4 1 2 + 1 3 4 µ+2− µ+1 µ+3− µ+2 − 1 µ+4− µ+3 + 1 µ+5− µ+4 1 1 2 2 3 3 4 4 µ+3− µ+2 µ+4− µ+3 µ+5− µ+4 − 1 µ+6− µ+5 + 1 1 1 2 2 3 3 4 4 µ+4 µ+3 µ+5 µ+4 µ+6 µ+5 µ+7 µ+6 1 − 1 2 − 2 3 − 3 4 − 4 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 0 0 0 −1 1 0 0 µ 0 1 0 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+2 µ+3 µ+4 1 2 + 1 3 4 µ+1 µ+2 − 1 µ+3 + 1 µ+4 0 1 2 3 µ+2 µ+3 µ+4 − 1 µ+5 + 1 0 1 2 3 µ+3 µ+4 µ+5 µ+6 0 1 2 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 0 0 −1 1 0 0 µ 0 1 0 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+2 µ+1 µ+3 µ+4 1 2 + 1 + 1 3 4 µ+1 µ+2+ µ+1 − 1 µ+3 + 1 µ+4 0 1 0 2 3 µ+2 µ+3+ µ+2 µ+4 − 1 µ+5 + 1 0 1 0 2 3 µ+3 µ+4 µ+3 µ+5 µ+6 0 1 + 0 2 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 0 0 −1 1 0 0 µ 0 1 0 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+2 µ+2 µ+3 µ+4 1 2 + 1 3 4 µ+1 µ+2+ µ+2 − 1 µ+3 + 1 µ+4 0 1 0 2 3 µ+2 µ+3+ µ+3 µ+4 − 1 µ+5 + 1 0 1 0 2 3 µ+3 µ+4 µ+4 µ+5 µ+6 0 1 + 0 2 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 0 0 −1 1 0 0 µ 0 1 0 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+3 µ+4 1 2 3 4 µ+1 µ+3 − 1 µ+3 + 1 µ+4 0 1 2 3 µ+2 µ+4 µ+4 − 1 µ+5 + 1 0 1 2 3 µ+3 µ+5 µ+5 µ+6 0 1 2 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 1 0 −1 1 0 0 µ 0 1 1 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+3 µ+3 µ+4 1 2 3 + 2 4 µ+1 µ+3 − 1 µ+3+ µ+3 µ+4 0 1 2 1 3 µ+2 µ+4 µ+4+ µ+4 − 1 µ+5 + 1 0 1 2 1 3 µ+3 µ+5 µ+5 µ+5 µ+6 0 1 2 + 1 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 1 0 −1 1 0 0 µ 0 1 1 0 ·D (4) · = 0 −1 1 0 2,1 0 0 1 0 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+4 µ+4 1 2 3 4 µ+1 µ+3 − 1 µ+4 µ+4 0 1 2 3 µ+2 µ+4 µ+5 − 1 µ+5 + 1 0 1 2 3 µ+3 µ+5 µ+6 µ+6 0 1 2 3 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 1 1 −1 1 0 0 µ 0 1 1 1 ·D (4) · = 0 −1 1 0 2,1 0 0 1 1 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+4 µ+4 µ+4 1 2 3 4 + 3 µ+1 µ+3 − 1 µ+4 µ+4+ µ+4 0 1 2 3 2 µ+2 µ+4 µ+5 − 1 µ+5+ µ+5 0 1 2 3 2 µ+3 µ+5 µ+6 µ+6 µ+6 0 1 2 3 + 2 − 1 15 / 32 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 Matrix Transformations 1 0 0 0 1 1 1 1 −1 1 0 0 µ 0 1 1 1 ·D (4) · = 0 −1 1 0 2,1 0 0 1 1 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+4 µ+5 1 2 3 4 µ+1 µ+3 − 1 µ+4 µ+5 0 1 2 3 µ+2 µ+4 µ+5 − 1 µ+6 0 1 2 3 µ+3 µ+5 µ+6 µ+7 0 1 2 3 − 1 15 / 32 Matrix Transformations 1 0 0 0 1 1 1 1 −1 1 0 0 µ 0 1 1 1 ·D (4) · = 0 −1 1 0 2,1 0 0 1 1 0 0 −1 1 0 0 0 1 µ+1 µ+3 µ+4 µ+5 1 2 3 4 µ+1 µ+3 − 1 µ+4 µ+5 0 1 2 3 µ+2 µ+4 µ+5 − 1 µ+6 0 1 2 3 µ+3 µ+5 µ+6 µ+7 0 1 2 3 − 1 ∗ ∗ ∗ ∗ µ 1 L·D2,1(4) ·R = µ+3 1 E1,1 (3) 1 15 / 32 ! µ = Rs,1(n). Laplace expansion: a˜1,1 a˜1,2 ··· a˜1,n µ 1 As,1(n) = det µ+3 1 Bs−1,1(n − 1) 1 =a ˜1,1 · Cof1,1(n − 1) + ... +a ˜1,n · Cof1,n(n − 1). With cn,j := Cof1,j(n − 1)/Cof1,1(n − 1), we obtain µ n As,1(n) X µ+3 = a˜1,j · cn,j Bs−1,1(n − 1) j=1 Proof via the Holonomic Ansatz To show: µ A (n) (n + s − 2)(µ − 1)(µ + n + 1)(µ + s) s,1 = µ+3 2n(s − 1)(µ + 2)(µ + n + s − 1) Bs−1,1(n − 1) | {zµ } =:Rs,1(n) 16 / 32 ! µ = Rs,1(n). With cn,j := Cof1,j(n − 1)/Cof1,1(n − 1), we obtain µ n As,1(n) X µ+3 = a˜1,j · cn,j Bs−1,1(n − 1) j=1 Proof via the Holonomic Ansatz To show: µ A (n) (n + s − 2)(µ − 1)(µ + n + 1)(µ + s) s,1 = µ+3 2n(s − 1)(µ + 2)(µ + n + s − 1) Bs−1,1(n − 1) | {zµ } =:Rs,1(n) Laplace expansion: a˜1,1 a˜1,2 ··· a˜1,n µ 1 As,1(n) = det µ+3 1 Bs−1,1(n − 1) 1 =a ˜1,1 · Cof1,1(n − 1) + ... +a ˜1,n · Cof1,n(n − 1). 16 / 32 ! µ = Rs,1(n). Proof via the Holonomic Ansatz To show: µ A (n) (n + s − 2)(µ − 1)(µ + n + 1)(µ + s) s,1 = µ+3 2n(s − 1)(µ + 2)(µ + n + s − 1) Bs−1,1(n − 1) | {zµ } =:Rs,1(n) Laplace expansion: a˜1,1 a˜1,2 ··· a˜1,n µ 1 As,1(n) = det µ+3 1 Bs−1,1(n − 1) 1 =a ˜1,1 · Cof1,1(n − 1) + ... +a ˜1,n · Cof1,n(n − 1). With cn,j := Cof1,j(n − 1)/Cof1,1(n − 1), we obtain µ n As,1(n) X µ+3 = a˜1,j · cn,j Bs−1,1(n − 1) j=1 16 / 32 Proof via the Holonomic Ansatz To show: µ A (n) (n + s − 2)(µ − 1)(µ + n + 1)(µ + s) s,1 = µ+3 2n(s − 1)(µ + 2)(µ + n + s − 1) Bs−1,1(n − 1) | {zµ } =:Rs,1(n) Laplace expansion: a˜1,1 a˜1,2 ··· a˜1,n µ 1 As,1(n) = det µ+3 1 Bs−1,1(n − 1) 1 =a ˜1,1 · Cof1,1(n − 1) + ... +a ˜1,n · Cof1,n(n − 1). With cn,j := Cof1,j(n − 1)/Cof1,1(n − 1), we obtain µ n As,1(n) X ! µ µ+3 = a˜1,j · cn,j = Rs,1(n). Bs−1,1(n − 1) j=1 16 / 32 These recurrences can be constructed from precomputed data via ansatz with undetermined coefficients and interpolation. [1] [1] [1] [1] p0,2 · cn,j+2 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [2] [2] [2] [2] p1,1 · cn+1,j+1 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] [3] [3] [3] p2,0 · cn+2,j + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] p2,0 = −(j − 2n − 4)(j − 2n − 3)(µ + 6n + 5)(µ + 6n + 7)(µ + 6n + 9)(n + r − 1)(n + r)(j + µ + 2n + 3)(j + µ + 2n + 4)(2j4 + 3j3µ − 6j3n + j3 + j2µ2 − 12j2µn − 3j2µ + 12j2n2 − 30j2n − 8j2 − 4jµ2n − 2jµ2 + 24jµn2 − 8jµn − 6jµ + 72jn2 + 12jn − 4j + 8µ2n2 + 4µ2n + 40µn2 + 20µn + 48n2 + 24n)(µ + 2n + 2r)(µ + 2n + 2r + 1)(µ + 2n + 2r + 2)(µ + 2n + 2r + 3)(µ + 4n + 2r + 1) Proof via the Holonomic Ansatz Guess: cn,j satisfies a holonomic system of recurrence equations. 17 / 32 [1] [1] [1] [1] p0,2 · cn,j+2 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [2] [2] [2] [2] p1,1 · cn+1,j+1 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] [3] [3] [3] p2,0 · cn+2,j + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] p2,0 = −(j − 2n − 4)(j − 2n − 3)(µ + 6n + 5)(µ + 6n + 7)(µ + 6n + 9)(n + r − 1)(n + r)(j + µ + 2n + 3)(j + µ + 2n + 4)(2j4 + 3j3µ − 6j3n + j3 + j2µ2 − 12j2µn − 3j2µ + 12j2n2 − 30j2n − 8j2 − 4jµ2n − 2jµ2 + 24jµn2 − 8jµn − 6jµ + 72jn2 + 12jn − 4j + 8µ2n2 + 4µ2n + 40µn2 + 20µn + 48n2 + 24n)(µ + 2n + 2r)(µ + 2n + 2r + 1)(µ + 2n + 2r + 2)(µ + 2n + 2r + 3)(µ + 4n + 2r + 1) Proof via the Holonomic Ansatz Guess: cn,j satisfies a holonomic system of recurrence equations. These recurrences can be constructed from precomputed data via ansatz with undetermined coefficients and interpolation. 17 / 32 [3] p2,0 = −(j − 2n − 4)(j − 2n − 3)(µ + 6n + 5)(µ + 6n + 7)(µ + 6n + 9)(n + r − 1)(n + r)(j + µ + 2n + 3)(j + µ + 2n + 4)(2j4 + 3j3µ − 6j3n + j3 + j2µ2 − 12j2µn − 3j2µ + 12j2n2 − 30j2n − 8j2 − 4jµ2n − 2jµ2 + 24jµn2 − 8jµn − 6jµ + 72jn2 + 12jn − 4j + 8µ2n2 + 4µ2n + 40µn2 + 20µn + 48n2 + 24n)(µ + 2n + 2r)(µ + 2n + 2r + 1)(µ + 2n + 2r + 2)(µ + 2n + 2r + 3)(µ + 4n + 2r + 1) Proof via the Holonomic Ansatz Guess: cn,j satisfies a holonomic system of recurrence equations. These recurrences can be constructed from precomputed data via ansatz with undetermined coefficients and interpolation. [1] [1] [1] [1] p0,2 · cn,j+2 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [2] [2] [2] [2] p1,1 · cn+1,j+1 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] [3] [3] [3] p2,0 · cn+2,j + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 17 / 32 Proof via the Holonomic Ansatz Guess: cn,j satisfies a holonomic system of recurrence equations. These recurrences can be constructed from precomputed data via ansatz with undetermined coefficients and interpolation. [1] [1] [1] [1] p0,2 · cn,j+2 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [2] [2] [2] [2] p1,1 · cn+1,j+1 + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] [3] [3] [3] p2,0 · cn+2,j + p1,0 · cn+1,j + p0,1 · cn,j+1 + p0,0 · cn,j = 0 [3] p2,0 = −(j − 2n − 4)(j − 2n − 3)(µ + 6n + 5)(µ + 6n + 7)(µ + 6n + 9)(n + r − 1)(n + r)(j + µ + 2n + 3)(j + µ + 2n + 4)(2j4 + 3j3µ − 6j3n + j3 + j2µ2 − 12j2µn − 3j2µ + 12j2n2 − 30j2n − 8j2 − 4jµ2n − 2jµ2 + 24jµn2 − 8jµn − 6jµ + 72jn2 + 12jn − 4j + 8µ2n2 + 4µ2n + 40µn2 + 20µn + 48n2 + 24n)(µ + 2n + 2r)(µ + 2n + 2r + 1)(µ + 2n + 2r + 2)(µ + 2n + 2r + 3)(µ + 4n + 2r + 1) 17 / 32 c2m,1 = 1, 2m X µ + i + j + 2r − 3 · c − c = 0, (2 i 2m), j − 1 2m,j 2m,i+2r−2 6 6 j=1 Cof1,j (n−1) I Abandon the original definition cn,j := . Cof1,1(n−1) I Use the conjectured holonomic description for cn,j instead. Cof1,j (n−1) I The first two identities prove c = . n,j Cof1,1(n−1) I The third identity proves the claimed quotient of determinants. Proof via the Holonomic Ansatz Prove: in the case where (A, B, s, n) is (D,E, 2r, 2m) 2m 2r−1 X µ + j + 2r − 1 X · c − c = Rµ (2m). j 2m,j 2m,j 2r,1 j=1 j=1 18 / 32 c2m,1 = 1, 2m X µ + i + j + 2r − 3 · c − c = 0, (2 i 2m), j − 1 2m,j 2m,i+2r−2 6 6 j=1 I Use the conjectured holonomic description for cn,j instead. Cof1,j (n−1) I The first two identities prove c = . n,j Cof1,1(n−1) I The third identity proves the claimed quotient of determinants. Proof via the Holonomic Ansatz Prove: in the case where (A, B, s, n) is (D,E, 2r, 2m) 2m 2r−1 X µ + j + 2r − 1 X · c − c = Rµ (2m). j 2m,j 2m,j 2r,1 j=1 j=1 Cof1,j (n−1) I Abandon the original definition cn,j := . Cof1,1(n−1) 18 / 32 c2m,1 = 1, 2m X µ + i + j + 2r − 3 · c − c = 0, (2 i 2m), j − 1 2m,j 2m,i+2r−2 6 6 j=1 Cof1,j (n−1) I The first two identities prove c = . n,j Cof1,1(n−1) I The third identity proves the claimed quotient of determinants. Proof via the Holonomic Ansatz Prove: in the case where (A, B, s, n) is (D,E, 2r, 2m) 2m 2r−1 X µ + j + 2r − 1 X · c − c = Rµ (2m). j 2m,j 2m,j 2r,1 j=1 j=1 Cof1,j (n−1) I Abandon the original definition cn,j := . Cof1,1(n−1) I Use the conjectured holonomic description for cn,j instead. 18 / 32 I The third identity proves the claimed quotient of determinants. Proof via the Holonomic Ansatz Prove: in the case where (A, B, s, n) is (D,E, 2r, 2m) c2m,1 = 1, 2m X µ + i + j + 2r − 3 · c − c = 0, (2 i 2m), j − 1 2m,j 2m,i+2r−2 6 6 j=1 2m 2r−1 X µ + j + 2r − 1 X · c − c = Rµ (2m). j 2m,j 2m,j 2r,1 j=1 j=1 Cof1,j (n−1) I Abandon the original definition cn,j := . Cof1,1(n−1) I Use the conjectured holonomic description for cn,j instead. Cof1,j (n−1) I The first two identities prove c = . n,j Cof1,1(n−1) 18 / 32 Proof via the Holonomic Ansatz Prove: in the case where (A, B, s, n) is (D,E, 2r, 2m) c2m,1 = 1, 2m X µ + i + j + 2r − 3 · c − c = 0, (2 i 2m), j − 1 2m,j 2m,i+2r−2 6 6 j=1 2m 2r−1 X µ + j + 2r − 1 X · c − c = Rµ (2m). j 2m,j 2m,j 2r,1 j=1 j=1 Cof1,j (n−1) I Abandon the original definition cn,j := . Cof1,1(n−1) I Use the conjectured holonomic description for cn,j instead. Cof1,j (n−1) I The first two identities prove c = . n,j Cof1,1(n−1) I The third identity proves the claimed quotient of determinants. 18 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 19 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 19 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 19 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 19 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 19 / 32 Lemma: For m, r ∈ Z and m > r > 1, 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim = , ε→0 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim = . ε→0 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) Proof of a Conjecture for s = −1 Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 Lemma: For m, r ∈ Z and m > r > 1, 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim = , ε→0 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim = . ε→0 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) Proof of a Conjecture for s = −1 µ D−1,2r(2m) µ+3 E−1,2r−1(2m − 1) µ E−1,2r+1(2m + 1) µ+3 D−1,2r(2m) Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 Lemma: For m, r ∈ Z and m > r > 1, 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim = , ε→0 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim = . ε→0 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) Proof of a Conjecture for s = −1 µ D2r,−1(2m) µ+3 E2r−1,−1(2m − 1) µ E2r+1,−1(2m + 1) µ+3 D2r,−1(2m) Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 Lemma: For m, r ∈ Z and m > r > 1, 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim , ε→0 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim . ε→0 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) Proof of a Conjecture for s = −1 µ D (2m) 0 2r,−1 = µ+3 0 E2r−1,−1(2m − 1) µ E (2m + 1) 0 2r+1,−1 = µ+3 0 D2r,−1(2m) Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 Lemma: For m, r ∈ Z and m > r > 1, 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim , ε→0 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim . ε→0 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) Proof of a Conjecture for s = −1 µ D2r+ε,−1+ε(2m) µ+3 = E2r−1+ε,−1+ε(2m − 1) µ E2r+1+ε,−1+ε(2m + 1) µ+3 = D2r+ε,−1+ε(2m) Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 Proof of a Conjecture for s = −1 Lemma: For m, r ∈ Z and m > r > 1, µ D (2m) 2r(2m − 1)(µ − 3)(µ + 2m + 2r − 2) lim 2r+ε,−1+ε = , ε→0 µ+3 µ(m + r)(µ + 2m − 3)(µ + 2r − 2) E2r−1+ε,−1+ε(2m − 1) µ E (2m + 1) 2m(2r + 1)(µ − 3)(µ + 2m + 2r) lim 2r+1+ε,−1+ε = . ε→0 µ+3 µ(m + r + 1)(µ + 2m − 2)(µ + 2r − 1) D2r+ε,−1+ε(2m) Theorem: For m, r ∈ Z and m > r > 1, µ E−1,2r−1(2m + 1) = m−r 2m ! (−1) (3 − µ)(m + r + 1)m−r Y (µ + i − 3)2r · 22m−2r+1 µ + r − 3 (i) 2 2 m−r+1 i=1 2r m−r 2 µ 2 ! Y µ + 2i + 6r − 3 2 + 2i + 3r − 1 × i i−1 . 2 µ 2 i=1 i i 2 + i + 3r − 1 i−1 20 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = row/column adapt 2nd ε = 0 Taylor LAR ˜ ˜tr operations column A truncation A det det det = linearity of = A A + O(ε2) A + O(ε2) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ As+ε,−1+ε(n) det µ As+ε,−1+ε(n) 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = adapt 2nd ε = 0 Taylor ˜ ˜tr column A truncation A det det det = linearity of = A A + O(ε2) A + O(ε2) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column As+ε,−1+ε(n) LAR operations det µ As+ε,−1+ε(n) 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = adapt 2nd ε = 0 Taylor ˜ ˜tr column A truncation A det det linearity of = A + O(ε2) A + O(ε2) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column As+ε,−1+ε(n) LAR operations det det µ = As+ε,−1+ε(n) A 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = ε = 0 Taylor ˜tr truncation A det det linearity of = A + O(ε2) A + O(ε2) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column adapt 2nd As+ε,−1+ε(n) LAR ˜ operations column A det det µ = As+ε,−1+ε(n) A 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = ε = 0 Taylor ˜tr truncation A det = A + O(ε2) mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column adapt 2nd As+ε,−1+ε(n) LAR ˜ operations column A det det det linearity of µ = 2 As+ε,−1+ε(n) A A + O(ε ) determinant 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = det = A + O(ε2) mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det linearity of µ = 2 As+ε,−1+ε(n) A A + O(ε ) determinant 21 / 32 n X = a˜ · c + O(ε) i,1 n,i Btr i=2 = Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Bs−1+ε,−1+ε(n − 1) µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 21 / 32 n X = a˜i,1 · cn,i + O(ε) i=2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Btr Bs−1+ε,−1+ε(n − 1) = µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 21 / 32 n X = a˜i,1 · cn,i + O(ε) i=2 ε · Cofi,1(n − 1) cn,i := lim ε→0 Cof1,1(n − 1) Proof Layout µ As+ε,−1+ε(n) µ+3 Btr Bs−1+ε,−1+ε(n − 1) = µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 Laplace n X a˜i,1 · Cofi,1(n − 1) i=2 21 / 32 n X = a˜i,1 · cn,i + O(ε) i=2 Proof Layout µ As+ε,−1+ε(n) µ+3 Btr Bs−1+ε,−1+ε(n − 1) = µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 21 / 32 Proof Layout µ n As+ε,−1+ε(n) X = a˜ · c + O(ε) µ+3 i,1 n,i Btr Bs−1+ε,−1+ε(n − 1) i=2 = µ row/column adapt 2nd ε = 0 Taylor tr As+ε,−1+ε(n) LAR ˜ ˜ operations column A truncation A det det det det linearity of µ = 2 = 2 As+ε,−1+ε(n) A A + O(ε ) A + O(ε ) determinant mod ε2 Laplace n ε · Cofi,1(n − 1) X cn,i := lim a˜i,1 · Cofi,1(n − 1) ε→0 Cof1,1(n − 1) i=2 21 / 32 µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε + O(ε2) ε + O(ε2) µ+i+j+s−5 ∓ δ + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 s,j−i (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) µ+3 Bs−1+ε,−1+ε(n − 1) = Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) 22 / 32 j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε + O(ε2) ε + O(ε2) µ+i+j+s−5 ∓ δ + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 s,j−i (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) µ+3 Bs−1+ε,−1+ε(n − 1) = Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) 22 / 32 µ+3 Bs−1+ε,−1+ε(n − 1) = Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε + O(ε2) ε + O(ε2) µ+i+j+s−5 ∓ δ + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 s,j−i (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) 22 / 32 Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε 2 ε 2 µ+i+j+s−5 + O(ε ) + O(ε ) ∓ δs,j−i + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) µ+i+j+s−3+2ε µ+3 j−2+ε ∓ δs,j−i Bs−1+ε,−1+ε(n − 1) = (1 6 i,j 6 n−1) 22 / 32 Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε 2 ε 2 µ+i+j+s−5 + O(ε ) + O(ε ) ∓ δs,j−i + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) ε 2 µ+i+j+s−3 µ+3 µ+i+s−1 + O(ε ) j−2 ∓ δs,j−i + O(ε) Bs−1+ε,−1+ε(n−1) = (1 6 i 6 n−1) (1 6 i 6 n−1,2 6 j 6 n−1) 22 / 32 Matrix Transformations µ+i+j+s−5+2ε µ j−2+ε ± δs,j−i−1 As+ε,−1+ε(n) = (1 6 i,j 6 n) µ Ln ·As+ε,−1+ε(n) ·Rn = j µ+s−3+2ε µ+j+s−3+2ε µ+s−3+2ε P −1+ε j−2+ε − −2+ε ± δs,k−2 k=1 (2 j n) 6 6 µ+i+s−5+2ε µ+i+j+s−5+2ε µ+i+s−5+2ε −2+ε j−3+ε − −3+ε ∓ δs,j−i (2 6 i 6 n) (2 6 i,j 6 n) j ε + O(ε2) 1 + O(ε) µ+j+s−3 ± P δ + O(ε) µ+s−2 j−2 s,k−2 k=1 (3 j n) 6 6 −ε 2 ε 2 µ+i+j+s−5 + O(ε ) + O(ε ) ∓ δs,j−i + O(ε) (µ+i+s−4)2 µ+i+s−2 j−3 (2 6 i 6 n) (2 6 i 6 n) (2 6 i 6 n, 3 6 j 6 n) ε 2 µ+i+j+s−5 µ+3 µ+i+s−2 + O(ε ) j−3 ∓ δs,j−i + O(ε) Bs−1+ε,−1+ε(n−1) = (2 6 i 6 n) (2 6 i 6 n,3 6 j 6 n) 22 / 32 n X 1 · c = −1 µ + i + s − 2 n,i i=2 n X µ + i + j + s − 5 · c = ±c (3 j n) j − 3 n,i n,j−s 6 6 i=2 ε · Cofi,1(n − 1) I Abandon the original definition cn,i := lim . ε→0 Cof1,1(n − 1) I Use the conjectured holonomic description for cn,i instead. ε · Cofi,1(n − 1) I The first two identities prove cn,i = lim . ε→0 Cof1,1(n − 1) I The third identity proves the claimed quotient of determinants. The Holonomic Ansatz n X −1 2s(n − 1)(µ − 3)(µ + n + s − 2) · c = (µ + i + s − 4) n,i µ(n + s)(µ + n − 3)(µ + s − 2) i=2 2 23 / 32 n X 1 · c = −1 µ + i + s − 2 n,i i=2 n X µ + i + j + s − 5 · c = ±c (3 j n) j − 3 n,i n,j−s 6 6 i=2 I Use the conjectured holonomic description for cn,i instead. ε · Cofi,1(n − 1) I The first two identities prove cn,i = lim . ε→0 Cof1,1(n − 1) I The third identity proves the claimed quotient of determinants. The Holonomic Ansatz n X −1 2s(n − 1)(µ − 3)(µ + n + s − 2) · c = (µ + i + s − 4) n,i µ(n + s)(µ + n − 3)(µ + s − 2) i=2 2 ε · Cofi,1(n − 1) I Abandon the original definition cn,i := lim . ε→0 Cof1,1(n − 1) 23 / 32 n X 1 · c = −1 µ + i + s − 2 n,i i=2 n X µ + i + j + s − 5 · c = ±c (3 j n) j − 3 n,i n,j−s 6 6 i=2 ε · Cofi,1(n − 1) I The first two identities prove cn,i = lim . ε→0 Cof1,1(n − 1) I The third identity proves the claimed quotient of determinants. The Holonomic Ansatz n X −1 2s(n − 1)(µ − 3)(µ + n + s − 2) · c = (µ + i + s − 4) n,i µ(n + s)(µ + n − 3)(µ + s − 2) i=2 2 ε · Cofi,1(n − 1) I Abandon the original definition cn,i := lim . ε→0 Cof1,1(n − 1) I Use the conjectured holonomic description for cn,i instead. 23 / 32 I The third identity proves the claimed quotient of determinants. The Holonomic Ansatz n X 1 · c = −1 µ + i + s − 2 n,i i=2 n X µ + i + j + s − 5 · c = ±c (3 j n) j − 3 n,i n,j−s 6 6 i=2 n X −1 2s(n − 1)(µ − 3)(µ + n + s − 2) · c = (µ + i + s − 4) n,i µ(n + s)(µ + n − 3)(µ + s − 2) i=2 2 ε · Cofi,1(n − 1) I Abandon the original definition cn,i := lim . ε→0 Cof1,1(n − 1) I Use the conjectured holonomic description for cn,i instead. ε · Cofi,1(n − 1) I The first two identities prove cn,i = lim . ε→0 Cof1,1(n − 1) 23 / 32 The Holonomic Ansatz n X 1 · c = −1 µ + i + s − 2 n,i i=2 n X µ + i + j + s − 5 · c = ±c (3 j n) j − 3 n,i n,j−s 6 6 i=2 n X −1 2s(n − 1)(µ − 3)(µ + n + s − 2) · c = (µ + i + s − 4) n,i µ(n + s)(µ + n − 3)(µ + s − 2) i=2 2 ε · Cofi,1(n − 1) I Abandon the original definition cn,i := lim . ε→0 Cof1,1(n − 1) I Use the conjectured holonomic description for cn,i instead. ε · Cofi,1(n − 1) I The first two identities prove cn,i = lim . ε→0 Cof1,1(n − 1) I The third identity proves the claimed quotient of determinants. 23 / 32 Calculations went close to the limits and required a lot of “human guidance”, for several reasons: I intermediate results are quite large (several 100 MB) due to the extra two parameters µ and r I some of the certificates had poles close to the summation boundaries I additional sums coming from the Kronecker deltas in combination with the row and column operations, some of which having non-natural boundaries Computational Challenge 24 / 32 I intermediate results are quite large (several 100 MB) due to the extra two parameters µ and r I some of the certificates had poles close to the summation boundaries I additional sums coming from the Kronecker deltas in combination with the row and column operations, some of which having non-natural boundaries Computational Challenge Calculations went close to the limits and required a lot of “human guidance”, for several reasons: 24 / 32 I some of the certificates had poles close to the summation boundaries I additional sums coming from the Kronecker deltas in combination with the row and column operations, some of which having non-natural boundaries Computational Challenge Calculations went close to the limits and required a lot of “human guidance”, for several reasons: I intermediate results are quite large (several 100 MB) due to the extra two parameters µ and r 24 / 32 I additional sums coming from the Kronecker deltas in combination with the row and column operations, some of which having non-natural boundaries Computational Challenge Calculations went close to the limits and required a lot of “human guidance”, for several reasons: I intermediate results are quite large (several 100 MB) due to the extra two parameters µ and r I some of the certificates had poles close to the summation boundaries 24 / 32 Computational Challenge Calculations went close to the limits and required a lot of “human guidance”, for several reasons: I intermediate results are quite large (several 100 MB) due to the extra two parameters µ and r I some of the certificates had poles close to the summation boundaries I additional sums coming from the Kronecker deltas in combination with the row and column operations, some of which having non-natural boundaries 24 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 25 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 25 / 32 µ µ Es,t(n) Family Ds,t(n) Family t t n odd n even s s t t n even n odd s s 25 / 32 µ µ + 2m + 4r − 1 m−r+1 2 + 2m + r + 1 m−r µ , m − r + 1 m−r+1 2 + m + 2r m−r m−r µ µ 1 (−1) 2 + 2m + r + 1 m−r 2 + 3r − 2 m−r+1 3 , 2 m−r m − r m−r m−r 1 (−1) 2 m−r+1 µ + 2m + 4r − 1 m−r+1 µ µ 1 . (2m − 2r + 1) 2 + m + 2r m−r 2 + 3r − 2 m−r+1 Triangle Relations Corollary: Let µ be an indeterminate, and let m, r ∈ Z. If m > r > 1, then µ E2r,1(2m + 1) µ = E2r,1(2m) µ E2r,1(2m + 1) µ = E2r+1,1(2m) µ E2r+1,1(2m) µ = E2r,1(2m) 26 / 32 Triangle Relations Corollary: Let µ be an indeterminate, and let m, r ∈ Z. If m > r > 1, then Eµ (2m + 1) µ + 2m + 4r − 1 µ + 2m + r + 1 2r,1 = m−r+1 2 m−r , Eµ (2m) µ 2r,1 m − r + 1 m−r+1 2 + m + 2r m−r Eµ (2m + 1) (−1)m−r µ + 2m + r + 1 µ + 3r − 1 2r,1 = 2 m−r 2 2 m−r+1 , Eµ (2m) 3 2r+1,1 2 m−r m − r m−r Eµ (2m) (−1)m−r 1 µ + 2m + 4r − 1 2r+1,1 = 2 m−r+1 m−r+1 . Eµ (2m) µ µ 1 2r,1 (2m − 2r + 1) 2 + m + 2r m−r 2 + 3r − 2 m−r+1 26 / 32 µ Conjecture for D1,1(n) 1 D1,1(n) = C(n) F (n) G 2 (n + 1) , where n i (−1)n + 3 Y ! C(n) = 2 , 2 i! i=1 b 3 b 1 (n−1)c−2c ! b 3 b n c−2c ! 2 2 2b 1 (i+2)c 2 2 2b 1 b n c− 1 (i−1)c−1 Y 3 Y 3 n 2 2 3 E(n) = (µ + 1)n µ + 2i + 6 · µ + 2i + 2 2 2 + 1 − 1 , i=1 i=1 1 n b 4 (n−1)c ! b 4 −1c ! Y 1−2i−m Y 1−2i−m Fm(n) = (µ + 2i + n + m) · (µ − 2i + 2n − 2m + 1) , i=1 i=1 E(n)F (n), if n is even, 0 1 (n−5) F (n) = 2 Y E(n)F1(n) (µ + 2i + 2n − 1), if n is odd, i=1 T (k) = 55296k6 + 41472(µ − 1)k5 + 384(30µ2 − 66µ + 53)k4 + 96(µ − 1)(15µ2 − 42µ + 61)k3 + 4(19µ4 − 122µ3 + 419µ2 − 544µ + 72)k2 + (µ − 1)(µ4 − 14µ3 + 101µ2 − 160µ − 84)k + 2(µ − 3)(µ − 2)(µ − 1)(µ + 1), n−1 6k 1 2 1 1 1 X 2 (µ + 8k − 1) 2 2k−1 2 (µ + 5) 2k−3 2 (µ + 4k + 2) k−2 2 (µ + 4k + 2) 2n−2k−2 S1(n) = · T (k), (2k)! 1 (µ + 6k − 3) k=1 2 3k+4 n−1 26k(µ + 8k + 3) 1 2 1 (µ + 5) 1 (µ + 4k + 4) 1 (µ + 4k + 4) X 2 2k 2 2k−2 2 k−2 2 2n−2k−2 1 S2(n) = · T k + , (2k + 1)! 1 (µ + 6k + 1) 2 k=1 2 3k+5 1 1 ! (µ + 6n − 3) (µ + 2) µ(µ − 1)S (n) P (n) = 23n−1 2 3n−2 2 2n−2 + 1 , 1 1 (µ + 3)2 213 2 (µ + 5) 2n−3 1 1 ! (µ + 6n + 1) (µ + 14) (µ + 4) µ(µ − 1)S (n) P (n) = 23n−1 2 3n−1 2 2n−2 + 2 , 2 1 (µ + 7)(µ + 9) 29 2 (µ + 5) 2n−2 ( P 1 (n + 1) , if n is odd, G(n) = 1 2 n P2 2 , if n is even. 27 / 32 µ Theorem for D1,1(n) 1 Let Dρk1,1(ben) = definedC(n) F (n as) Gρ02((na,+ b 1)) =, wherea and ρk(a, b) = b for k > 0. n i (−1)n + 3 Y ! If n isC an(n) odd= positive integer2 , then 2 i! i=1 3 1 3 n 2 (n+1)/2 b 2 b 2 (n−1)c−2c ! b 2 b 2 c−2c µ ! 1 2b 1 (i+2) c k−1 2b 1 b n c− 1 (i−1)c−1 Y 1 3 µ − 1 Y 3µ n+ 2j + 1 2 2 23 + 2j + 2 µ E(n) =X (µ + 1)n µ + 2i + 6 · 3k−2 µ + 2iY+ 2 + 1 − 1 j−1 , j−1 D (n) = ρk 4(µ − 2), 2 2 1,1 i=1 (2k − 1)! 2 µ + k −i=11 j µ + j + 1 k=01 n 2 2 k−1 j=1 j−1 2 2 j−1 b 4 (n−1)c ! b 4 −1c ! Y Y F (n) = (µ + 2i + n + m)1−2i−m · (µ − 2i + 2n − 2m + 1)1−2i−m , m (n−1)/2 µ + 2j2 µ + 2j − 1 µ + 2j + 3 i=1 Y j 2i=1 2 j 2 2 j+1 × . E(n)F0(n), if n is even, µ 1 2 1 j j + 1 + j + (n−j5)=k j j+1 2 2 j F (n) = 2 Y E(n)F1(n) (µ + 2i + 2n − 1), if n is odd, If n is an even positivei=1 integer then T (k) = 55296k6 + 41472(µ − 1)k5 + 384(30µ2 − 66µ + 53)k4 + 96(µ − 1)(15µ2 − 42µ + 61)k3 + 4(19µ4 − 122µ3 + 419µ2 − 544µ + 72)k2 + (µ − 1)(µ4 − 14µ3 + 101µ2 − 160µ − 84)k 2 n/2 k−1 µ + 2j + 1 µ + 2j + 1 X+ 2(µ − 3)(µ − 2)(µ − 1)(µ1+ 1), µ − 1 Y j−1 2 2 j−1 Dµ (n) = ρ 4(µ − 2), 3k−2 1,1 k 2 µ 1 µ 1 n−1 26k(µ + 8k − 1) 1 (2k −1 (µ1)!+ 5)2 1+(µk+− 4k + 2) 1 (µ + 4k + 2) j + j + kX=0 2 2k−1 2 2k−32 2 2 k−k−12 2 j=1 2n−2jk−−12 2 2 j−1 S1(n) = 1 · T (k), (2k)! 2 (µ + 6k − 3) 3k+4 k=1 n/2 µ 1 n/2−1 µ 3 n−1 26k(µ + 8k + 3)µ +1 2 2j 1 (µ + 5)+ 2j +1 (µ + 4k + 4) 1 (µ +µ 4k+ 4) 2j + 2j + X Y 2 2k 2j 2 2k−2 2 2 j−1 k−2Y2 2nj−2k−22 1 2 j+1 S2(n) = × · T k + , . µ(2k + 1)! 11(µ+ 6k + 1) µ 2 1 j + j + 2 3k+5 j + 1 + j + k=1 j=k j 2 2 j−1 j=k j+1 2 2 j 1 1 ! (µ + 6n − 3) (µ + 2) µ(µ − 1)S (n) P (n) = 23n−1 2 3n−2 2 2n−2 + 1 , 1 1 (µ + 3)2 213 2 (µ + 5) 2n−3 1 1 ! (µ + 6n + 1) (µ + 14) (µ + 4) µ(µ − 1)S (n) P (n) = 23n−1 2 3n−1 2 2n−2 + 2 , 2 1 (µ + 7)(µ + 9) 29 2 (µ + 5) 2n−2 ( P 1 (n + 1) , if n is odd, G(n) = 1 2 n P2 2 , if n is even. 28 / 32 I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: 29 / 32 I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas 29 / 32 I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities 29 / 32 I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants 29 / 32 I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) 29 / 32 I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions 29 / 32 Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) 29 / 32 Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. 29 / 32 (can be extended to recurrences, q-difference equations, etc.) The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). 29 / 32 The Holonomic Viewpoint The holonomic systems approach (Zeilberger 1990) is a versatile toolbox for solving many different kinds of mathematical problems: I calculate integrals and summation formulas I prove special function identities I evaluate symbolic determinants I computations in q-calculus (e.g., quantum knot invariants) I fast numerical evaluation of mathematical functions I number theory (irrationality proofs) Definition: A holonomic system is a maximally overdetermined system of linear partial differential equations. Algebraically, this corresponds to a left ideal in the Weyl algebra that has smallest possible dimension (Bernstein’s inequality). (can be extended to recurrences, q-difference equations, etc.) 29 / 32 I Recurrence equation: 2(ν − 1) J (x) = J (x) − J (x) ν x ν−1 ν−2 Many special functions can be characterized as solutions to systems of linear differential equations and recurrences, and in fact are holonomic. I Bessel differential equation: d2 d x2 J (x) + x J (x) + x2 − ν2J (x) = 0 dx2 ν dx ν ν Differential Equations and Recurrences Example: The Bessel function Jν(x) describes the vibrations of a circular membrane, as well as many other phenomena with cylindrical symmetry. 30 / 32 I Recurrence equation: 2(ν − 1) J (x) = J (x) − J (x) ν x ν−1 ν−2 Many special functions can be characterized as solutions to systems of linear differential equations and recurrences, and in fact are holonomic. Differential Equations and Recurrences Example: The Bessel function Jν(x) describes the vibrations of a circular membrane, as well as many other phenomena with cylindrical symmetry. I Bessel differential equation: d2 d x2 J (x) + x J (x) + x2 − ν2J (x) = 0 dx2 ν dx ν ν 30 / 32 Many special functions can be characterized as solutions to systems of linear differential equations and recurrences, and in fact are holonomic. Differential Equations and Recurrences Example: The Bessel function Jν(x) describes the vibrations of a circular membrane, as well as many other phenomena with cylindrical symmetry. I Bessel differential equation: d2 d x2 J (x) + x J (x) + x2 − ν2J (x) = 0 dx2 ν dx ν ν I Recurrence equation: 2(ν − 1) J (x) = J (x) − J (x) ν x ν−1 ν−2 30 / 32 Differential Equations and Recurrences Example: The Bessel function Jν(x) describes the vibrations of a circular membrane, as well as many other phenomena with cylindrical symmetry. I Bessel differential equation: d2 d x2 J (x) + x J (x) + x2 − ν2J (x) = 0 dx2 ν dx ν ν I Recurrence equation: 2(ν − 1) J (x) = J (x) − J (x) ν x ν−1 ν−2 Many special functions can be characterized as solutions to systems of linear differential equations and recurrences, and in fact are holonomic. 30 / 32 Table of Integrals by Gradshteyn and Ryzhik 31 / 32 Table of Integrals by Gradshteyn and Ryzhik 31 / 32 Table of Integrals by Gradshteyn and Ryzhik 31 / 32 Table of Integrals by Gradshteyn and Ryzhik 31 / 32 Table of Integrals by Gradshteyn and Ryzhik 31 / 32 I A large portion of such identities can be proven via the holonomic systems approach. I Algorithms are implemented in the HolonomicFunctions package. Gegenbauer Gamma Bessel (α) polynomials Cn (x) function Γ(x) function Jν(x) Table of Integrals by Gradshteyn and Ryzhik 31 / 32 Gamma Bessel function Γ(x) function Jν(x) I A large portion of such identities can be proven via the holonomic systems approach. I Algorithms are implemented in the HolonomicFunctions package. Table of Integrals by Gradshteyn and Ryzhik Gegenbauer (α) polynomials Cn (x) 31 / 32 Bessel function Jν(x) I A large portion of such identities can be proven via the holonomic systems approach. I Algorithms are implemented in the HolonomicFunctions package. Table of Integrals by Gradshteyn and Ryzhik Gegenbauer Gamma (α) polynomials Cn (x) function Γ(x) 31 / 32 I A large portion of such identities can be proven via the holonomic systems approach. I Algorithms are implemented in the HolonomicFunctions package. Table of Integrals by Gradshteyn and Ryzhik Gegenbauer Gamma Bessel (α) polynomials Cn (x) function Γ(x) function Jν(x) 31 / 32 I Algorithms are implemented in the HolonomicFunctions package. Table of Integrals by Gradshteyn and Ryzhik Gegenbauer Gamma Bessel (α) polynomials Cn (x) function Γ(x) function Jν(x) I A large portion of such identities can be proven via the holonomic systems approach. 31 / 32 Table of Integrals by Gradshteyn and Ryzhik Gegenbauer Gamma Bessel (α) polynomials Cn (x) function Γ(x) function Jν(x) I A large portion of such identities can be proven via the holonomic systems approach. I Algorithms are implemented in the HolonomicFunctions package. 31 / 32 I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. 32 / 32 I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. 32 / 32 I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences 32 / 32 I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties 32 / 32 I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping 32 / 32 I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) 32 / 32 I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras 32 / 32 Special function identities is one application area, but there are also many holonomic functions without a name! The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations 32 / 32 The HolonomicFunctions Package Example: Holonomic system, satisfied by both sides of the identity: 0 ia(n + 2ν)fn(a) + a(n + 1)fn+1(a) − in(n + 2ν)fn(a) = 0, a(n + 1)(n + 2)fn+2(a) − 2i(n + 1)(n + ν + 1)(n + 2ν + 1)fn+1(a) − a(n + 2ν)(n + 2ν + 1)fn(a) = 0. The HolonomicFunctions package is written in Mathematica, has been developed since 2007, and consists of ∼12,000 lines of code. I deals with multivariate holonomic functions and sequences I computes annihilating ideals by executing closure properties I summation and integration is done via creative telescoping I arithmetic in Ore algebras (differential-difference operators) I noncommutative Gr¨obnerbases in such operator algebras I coupled linear systems of differential-difference equations Special function identities is one application area, but there are also many holonomic functions without a name! 32 / 32