Rogers–Ramanujan type identities

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Rogers–Ramanujan type identities

Andrew Sills

Georgia Southern University

Seminar for Kyoto University October 6, 2020

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L. Euler (1707–1783)

C. G. J. Jacobi (1804–1851)

L. J. Rogers (1862–1933)

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Precursors to the RR identities

Throughout: Assume|q|<1.

X

n≥0

qⁿ

(1−q)(1−q²)· · ·(1−qⁿ) =

∞

Y

m=1

1

1−q^m (Euler)

X

n≥0

qⁿ²

(1−q)²(1−q²)²· · ·(1−qⁿ)² = Y

m≥1

1

1−q^m (Jacobi) X

n≥0

qⁿ²

(1−q)(1−q²)· · ·(1−qⁿ) = Y

m≥1 m≡±1(mod 5)

1 1−q^m

(Rogers)

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Precursors to the RR identities

X

n≥0

qⁿ

(1−q)(1−q²)· · ·(1−qⁿ) =

∞

Y

m=1

1

1−q^m (Euler)

X

n≥0

qⁿ²

(1−q)²(1−q²)²· · ·(1−qⁿ)² = Y

m≥1

1

1−q^m (Jacobi) X

n≥0

qⁿ²

(1−q)(1−q²)· · ·(1−qⁿ) = Y

m≥1 m≡±1(mod 5)

1 1−q^m

(Rogers)

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Precursors to the RR identities

X

n≥0

qⁿ

(1−q)(1−q²)· · ·(1−qⁿ) =

∞

Y

m=1

1

1−q^m (Euler)

X

n≥0

qⁿ²

(1−q)²(1−q²)²· · ·(1−qⁿ)² = Y

m≥1

1

1−q^m (Jacobi)

X

n≥0

qⁿ²

(1−q)(1−q²)· · ·(1−qⁿ) = Y

m≥1 m≡±1(mod 5)

1 1−q^m

(Rogers)

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Precursors to the RR identities

X

n≥0

qⁿ

(1−q)(1−q²)· · ·(1−qⁿ) =

∞

Y

m=1

1

1−q^m (Euler)

X

n≥0

qⁿ²

(1−q)²(1−q²)²· · ·(1−qⁿ)² = Y

m≥1

1

1−q^m (Jacobi) X

n≥0

qⁿ²

(1−q)(1−q²)· · ·(1−qⁿ) = Y

m≥1 m≡±1(mod 5)

1 1−q^m

(Rogers)

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Rising q-factorial notation

(a)n= (a;q)n:= (1−a)(1−aq)(1−aq²)· · ·(1−aqⁿ⁻¹),

(a)∞= (a;q)∞:= (1−a)(1−aq)(1−aq²)· · · ,

(a₁,a₂, . . .a_r;q)∞:= (a₁)∞(a₂)∞(a₃)∞· · ·(a_r)∞

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Rising q-factorial notation

(a)n= (a;q)n:= (1−a)(1−aq)(1−aq²)· · ·(1−aqⁿ⁻¹),

(a)∞= (a;q)∞:= (1−a)(1−aq)(1−aq²)· · ·,

(a₁,a₂, . . .a_r;q)∞:= (a₁)∞(a₂)∞(a₃)∞· · ·(a_r)∞

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Rising q-factorial notation

(a)n= (a;q)n:= (1−a)(1−aq)(1−aq²)· · ·(1−aqⁿ⁻¹),

(a)∞= (a;q)∞:= (1−a)(1−aq)(1−aq²)· · ·,

(a₁,a₂, . . .a_r;q)∞:= (a₁)∞(a₂)∞(a₃)∞· · ·(a_r)∞

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S. Ramanujan (1887–1920)

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Ramanujan’s “theta” function

For|ab|<1,

f(a,b) :=X

n∈Z

a^n(n+1)/2b^n(n−1)/2.

Jacobi’s triple product identity

f(a,b) = (a,b,ab;ab)∞.

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Ramanujan’s “theta” function

For|ab|<1,

f(a,b) :=X

n∈Z

a^n(n+1)/2b^n(n−1)/2.

Jacobi’s triple product identity

f(a,b) = (a,b,ab;ab)∞.

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Ramanujan’s notation

f(−q) :=f(−q,−q²) =X

n∈Z

(−1)ⁿq^n(3n−1)/2= (q)∞

(Euler’s pentagonal numbers thm)

ϕ(−q) :=f(−q,−q) =X

n∈Z

(−1)ⁿqⁿ² = (q)∞

(−q)∞

(Gauss’s square numbers thm)

ψ(−q) :=f(−q,−q³) =X

n∈Z

(−1)ⁿq^n(2n−1)= (q²;q²)∞

(−q;q²)∞

(Gauss’s hexagonal numbers thm)

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Ramanujan’s notation

f(−q) :=f(−q,−q²) =X

n∈Z

(−1)ⁿq^n(3n−1)/2= (q)∞

ϕ(−q) :=f(−q,−q) =X

n∈Z

(−1)ⁿqⁿ² = (q)∞

(−q)∞

ψ(−q) :=f(−q,−q³) =X

n∈Z

(−1)ⁿq^n(2n−1)= (q²;q²)∞

(−q;q²)∞

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Ramanujan’s notation

f(−q) :=f(−q,−q²) =X

n∈Z

(−1)ⁿq^n(3n−1)/2= (q)∞

ϕ(−q) :=f(−q,−q) =X

n∈Z

(−1)ⁿqⁿ² = (q)∞

(−q)∞

ψ(−q) :=f(−q,−q³) =X

n∈Z

(−1)ⁿq^n(2n−1)= (q²;q²)∞

(−q;q²)∞

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Rogers–Ramanujan identities

X

n≥0

qⁿ²

(q)_n = f(−q²,−q³) (q)_∞ .

X

n≥0

qⁿ⁽ⁿ⁺¹⁾

(q)_n = f(−q,−q⁴) (q)_∞ .

Ramanujan really enjoyed identities of this type. Over 50 are recorded in the lost notebook.

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Rogers–Ramanujan identities

X

n≥0

qⁿ²

(q)_n = f(−q²,−q³) (q)_∞ .

X

n≥0

qⁿ⁽ⁿ⁺¹⁾

(q)_n = f(−q,−q⁴) (q)_∞ . Ramanujan really enjoyed identities of this type.

Over 50 are recorded in the lost notebook.

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Rogers–Ramanujan identities

X

n≥0

qⁿ²

(q)_n = f(−q²,−q³) (q)_∞ .

X

n≥0

qⁿ⁽ⁿ⁺¹⁾

(q)_n = f(−q,−q⁴) (q)_∞ . Ramanujan really enjoyed identities of this type.

Over 50 are recorded in the lost notebook.

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Bailey pairs, Bailey’s lemma

If(α_n(a,q), β_n(a,q))satisfies

βn=

n

X

r=0

αr

(q)n−r(aq)n+r

,

then(αn, βn)is called aBailey pair with respect to a,

and (α⁰_n(a,q), β⁰_n(a,q))is also a Bailey pair, where

α⁰_r(a,q) = (ρ1)r(ρ2)r

(aq/ρ₁)r(aq/ρ₂)r

aq ρ₁ρ₂

r

αr

and

β_n⁰(a,q) =

n

X

j=0

(ρ1)j(ρ2)j(aq/ρ₁ρ2)n−j

(aq/ρ₁)n(aq/ρ₂)n(q)_n−j aq

ρ1ρ2

j

βj(a,q).

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Bailey pairs, Bailey’s lemma

If(α_n(a,q), β_n(a,q))satisfies

βn=

n

X

r=0

αr

(q)n−r(aq)n+r

,

then(αn, βn)is called aBailey pair with respect to a, and (α⁰_n(a,q), β⁰_n(a,q))is also a Bailey pair, where

α⁰_r(a,q) = (ρ1)r(ρ2)r

(aq/ρ₁)r(aq/ρ₂)r

aq ρ₁ρ₂

r

αr

and

β_n⁰(a,q) =

n

X

j=0

(ρ₁)_j(ρ₂)_j(aq/ρ₁ρ₂)n−j

(aq/ρ₁)n(aq/ρ₂)n(q)_n−j aq

ρ₁ρ₂ j

βj(a,q).

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Limiting cases of Bailey’s lemma

X

n≥0

qⁿ²βn(1,q) = 1 (q)∞

X

r≥0

q^r²αr(1,q) (PBL) X

n≥0

qⁿ²(−q;q²)_nβ_n(1,q²) = 1 ψ(−q)

X

r≥0

q^r²α_r(1,q²) (HBL)

X

n≥0

q^n(n+1)/2(−1)_nβn(1,q) = 2 ϕ(−q)

X

r≥0

q^r^(r^+1)/2

1+q^r αr(1,q) (SBL)

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Bailey, Dyson, and Slater

In the 1940’s, Bailey found a number of examples of Bailey pairs, and used them to generate RR type identities.

Freeman Dyson contributed a number of RR type identities to Bailey’s papers.

Lucy Slater found many Bailey pairs, and used them to generate a list of 130 RR type identities.

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Bailey, Dyson, and Slater

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Bailey, Dyson, and Slater

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General Bailey pairs

Ford |n, define

α^(d,e,k_n ⁾(a,q) := (−1)^n/da^(k/d−1)n/eq^(k/d−1+1/2d)n²/e−n/2e

(1−a^1/e)(q^d/e;q^d/e)_n/d ,

×(1−a^1/eq^2n/e)(a^1/e;q^d/e)_n/d,

˜

α^(d,e,k)_n (a,q) :=q^n(d−n)/2dea^−n/de(−a^1/e;q^d/e)_n/d

(−q^d/e;q^d/e)_n/dα^(d,e,k)_n (a,q),

¯

α^(d,e,k)_n (a,q) := (−1)^n/dqⁿ²^/2de (q^d/2e;q^d/e)_n/d

(a^1/eq^d/2e;q^d/e)_n/dα^(d,e,k)_n (a,q). Let the correspondingβ^(d,e,k)_n (a,q),β˜_n^(d,e,k)(a,q), and

β¯_n^(d,e,k⁾(a,q)be determined by the Bailey pair relation.

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General Bailey pairs

Ford |n, define

(1−a^1/e)(q^d/e;q^d/e)_n/d ,

×(1−a^1/eq^2n/e)(a^1/e;q^d/e)_n/d,

˜

(−q^d/e;q^d/e)_n/dα^(d,e,k_n ⁾(a,q),

¯

(a^1/eq^d/2e;q^d/e)_n/dα^(d,e,k)_n (a,q). Let the correspondingβ^(d,e,k)_n (a,q),β˜_n^(d,e,k)(a,q), and

β¯_n^(d,e,k⁾(a,q)be determined by the Bailey pair relation.

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General Bailey pairs

Ford |n, define

(1−a^1/e)(q^d/e;q^d/e)_n/d ,

×(1−a^1/eq^2n/e)(a^1/e;q^d/e)_n/d,

˜

¯

(a^1/eq^d/2e;q^d/e)_n/dα^(d,e,k_n ⁾(a,q).

Let the correspondingβ^(d,e,k)_n (a,q),β˜_n^(d,e,k)(a,q), and β¯_n^(d,e,k⁾(a,q)be determined by the Bailey pair relation.

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General Bailey pairs

Ford |n, define

(1−a^1/e)(q^d/e;q^d/e)_n/d ,

×(1−a^1/eq^2n/e)(a^1/e;q^d/e)_n/d,

˜

¯

(a^1/eq^d/2e;q^d/e)_n/dα^(d,e,k_n ⁾(a,q).

Let the correspondingβ^(d,e,k)_n (a,q),β˜_n^(d,e,k)(a,q), and β¯_n^(d,e,k⁾(a,q)be determined by the Bailey pair relation.

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For any positive integer triples(d,e,k), upon inserting any of theseα’s into any of the limiting cases of Bailey’s lemma witha=1, the resulting series is summable via Jacobi’s triple product identity.

For certain(d,e,k), the resulting expression forβis a very well-poised₆φ5, summable by a theorem of F. H. Jackson. Using only this, and an associated families ofq-difference equations, one can recover the majority of Slater’s list, as well as other identities.

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For certain(d,e,k), the resulting expression forβ is a very well-poised₆φ5, summable by a theorem of F. H. Jackson.

Using only this, and an associated families ofq-difference equations, one can recover the majority of Slater’s list, as well as other identities.

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For certain(d,e,k), the resulting expression forβ is a very well-poised₆φ5, summable by a theorem of F. H. Jackson.

Using only this, and an associated families ofq-difference equations, one can recover the majority of Slater’s list, as well as other identities.

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The Bailey pair that arises from

α^(1,1,2)_n (a,q), β_n^(1,1,2)(a,q)

= (−1)ⁿaⁿq^n(3n−1)/2(1−aq²ⁿ)(a)_n (1−a)(q)n

, 1 (q)n

!

yields

P

n≥0 qⁿ²

(q)n = ^f^(−q_(q)²^,−q³⁾

∞ upon insertion into (PBL), P

n≥0

qⁿ⁽ⁿ⁺¹⁾(−1)n

(q)n = ^ϕ(−q_ϕ(−q)²⁾ upon insertion into (SBL), and P

n≥0

qⁿ²(−q;q²)n

(q²;q²)n = ^f^(−q_ψ(−q)³^,−q⁵⁾ upon insertion into (HBL).

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α^(1,1,2)_n (a,q), β_n^(1,1,2)(a,q)

= (−1)ⁿaⁿq^n(3n−1)/2(1−aq²ⁿ)(a)_n (1−a)(q)n

, 1 (q)n

!

yields P

n≥0 qⁿ²

(q)n = ^f^(−q_(q)²^,−q³⁾

∞ upon insertion into (PBL),

P

n≥0

qⁿ⁽ⁿ⁺¹⁾(−1)n

n≥0

qⁿ²(−q;q²)n

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α^(1,1,2)_n (a,q), β_n^(1,1,2)(a,q)

= (−1)ⁿaⁿq^n(3n−1)/2(1−aq²ⁿ)(a)_n (1−a)(q)n

, 1 (q)n

!

yields P

n≥0 qⁿ²

(q)n = ^f^(−q_(q)²^,−q³⁾

n≥0

qⁿ⁽ⁿ⁺¹⁾(−1)n

(q)n = ^ϕ(−q_ϕ(−q)²⁾ upon insertion into (SBL), and

P

n≥0

qⁿ²(−q;q²)n

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α^(1,1,2)_n (a,q), β_n^(1,1,2)(a,q)

= (−1)ⁿaⁿq^n(3n−1)/2(1−aq²ⁿ)(a)_n (1−a)(q)n

, 1 (q)n

!

yields P

n≥0 qⁿ²

(q)n = ^f^(−q_(q)²^,−q³⁾

n≥0

qⁿ⁽ⁿ⁺¹⁾(−1)n

n≥0

qⁿ²(−q;q²)n

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New identities arising from this framework (S.)

X

n,r≥0

qⁿ²^+2nr+2r²(−q;q²)_r

(q)_2r(q)_n = f(−q¹⁰,−q¹⁰) (q)∞

by insertion of( ˜α^(2,1,5)_n (1,q),β˜_n^(2,1,5)(1,q))into (PBL).

X

n,r≥0

q⁴ⁿ²^+8nr+8r²(−q;q²)_2r

(q⁴;q⁴)_2r(q⁴;q⁴)_n = f(q⁹,q¹¹) (q⁴;q⁴)∞

by insertion of( ¯α^(1,2,4)_n (1,q),β¯_n^(1,2,4)(1,q))into (PBL).

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New identities arising from this framework (S.)

X

n,r≥0

qⁿ²^+2nr+2r²(−q;q²)_r

(q)_2r(q)_n = f(−q¹⁰,−q¹⁰) (q)∞

X

n,r≥0

q⁴ⁿ²^+8nr+8r²(−q;q²)_2r

(q⁴;q⁴)_2r(q⁴;q⁴)_n = f(q⁹,q¹¹) (q⁴;q⁴)∞

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New identities arising from this framework (S.)

X

n,r≥0

qⁿ²^+2nr+2r²(−q;q²)_r

(q)_2r(q)_n = f(−q¹⁰,−q¹⁰) (q)∞

X

n,r≥0

q⁴ⁿ²^+8nr+8r²(−q;q²)_2r

(q⁴;q⁴)_2r(q⁴;q⁴)_n = f(q⁹,q¹¹) (q⁴;q⁴)∞

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A family of mod 24 identities

X

n≥0

qⁿ⁽ⁿ⁺²⁾(−q;q²)n(−1;q⁶)n

(q²;q²)2n(−1;q²)n

= f(−q,−q¹¹)f(−q¹⁰,−q¹⁴) ψ(−q)(q²⁴;q²⁴)∞

(McLaughlin.-S.)

∞

X

n=0

qⁿ²(−q³;q⁶)_n

(q²;q²)_2n = f(−q²,−q¹⁰)f(−q⁸,−q¹⁶) ψ(−q)(q²⁴;q²⁴)∞

(Ramanujan)

X

n≥0

qⁿ²(−q;q²)_n(−1;q⁶)_n (q²;q²)_2n(−1;q²)n

= f(−q³,−q⁹)f(−q⁶,−q¹⁸) ψ(−q)(q²⁴;q²⁴)∞

(M.-S.)

X

n≥0

qⁿ⁽ⁿ⁺²⁾(−q³;q⁶)n

(q²;q²)_2n(1−q²ⁿ⁺¹) = f(−q⁴,−q⁸)f(−q⁴,−q²⁰) ψ(−q)(q²⁴;q²⁴)∞

(M.-S.)

qⁿ⁽ⁿ⁺²⁾(−q;q²) (−q⁶;q⁶) f(−q⁵,−q⁷)f(−q²,−q²²)

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Combinatorial considerations

Rogers, Ramanujan, Bailey, and Slater did not consider the combinatorial aspect of their work.

Apartitionλofnis a tuple(λ₁, λ₂, . . . , λ_l)of weakly decreasing positive integers (called thepartsofλ) that sum ton. The seven partitions of 5 are

(5),(4,1),(3,2),(3,1,1),(2,2,1),(2,1,1,1),(1,1,1,1,1).

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Combinatorial considerations

Apartitionλofnis a tuple(λ₁, λ₂, . . . , λ_l)of weakly decreasing positive integers (called thepartsofλ) that sum ton.

The seven partitions of 5 are

(5),(4,1),(3,2),(3,1,1),(2,2,1),(2,1,1,1),(1,1,1,1,1).

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Combinatorial considerations

Apartitionλofnis a tuple(λ₁, λ₂, . . . , λ_l)of weakly decreasing positive integers (called thepartsofλ) that sum ton. The seven partitions of 5 are

(5),(4,1),(3,2),(3,1,1),(2,2,1),(2,1,1,1),(1,1,1,1,1).

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Euler’s partition theorem

The number of partitions ofninto odd parts equals the number of partitions ofninto distinct parts.

Example:

9,711,531,51111,333,33111,3111111,111111111

9,81,72,63,621,54,531,432

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Euler’s partition theorem

Example:

9,711,531,51111,333,33111,3111111,111111111

9,81,72,63,621,54,531,432

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Euler’s partition theorem

Example:

9,711,531,51111,333,33111,3111111,111111111

9,81,72,63,621,54,531,432

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Combinatorial Rogers–Ramanujan (due to MacMahon and Schur)

The number of partitions ofninto parts that mutually differ by at least 2 equals the number of partitions ofninto parts congruent to±1 (mod 5).

The number of partitions ofninto parts greater than 1 that mutually differ by at least 2 equals the number of partitions ofn into parts congruent to±2 (mod 5).

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Combinatorial Rogers–Ramanujan (due to MacMahon and Schur)

The number of partitions ofninto parts that mutually differ by at least 2 equals the number of partitions ofninto parts congruent to±1 (mod 5).

The number of partitions ofninto parts greater than 1 that

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B. Gordon’s combinatorial generalization of RR (1961)

Letk be a positive integer and 1≤i≤k.

LetA_k,i(n)denote the number of partitions ofninto parts 6≡0,±i (mod 2k+1).

LetB_k,i(n)denote the number of partitionsλofnwhere at mosti−1 of the parts ofλequal 1,

λj−λj+k−1≥2 forj =1,2, . . . ,l(λ) +1−k. ThenA_k,i(n) =B_k,i(n)for alln.

Note:The casek =2 gives the standard combinatorial interpretation of the two RR identities.

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B. Gordon’s combinatorial generalization of RR (1961)

Letk be a positive integer and 1≤i≤k. LetA_k,i(n)denote the number of partitions ofninto parts 6≡0,±i (mod 2k+1).

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B. Gordon’s combinatorial generalization of RR (1961)

λj−λj+k−1≥2 forj =1,2, . . . ,l(λ) +1−k.

ThenA_k,i(n) =B_k,i(n)for alln.

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B. Gordon’s combinatorial generalization of RR (1961)

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B. Gordon’s combinatorial generalization of RR (1961)

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G. Andrews’ analytic counterpart to Gordon’s theorem

X

nk−1≥nk−2≥···≥n₁≥0

qⁿ²¹⁺ⁿ²²^+···+n²^k−1⁺ⁿⁱ⁺ⁿⁱ⁺¹^+···+n^k⁻¹ (q)_n₁(q)_n₂−n₁(q)_n₃−n₂· · ·(q)_n_k₋₁−n_k₋₂

= f(−qⁱ,−q^2k+1−i) (q)∞

.

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Combinatorial interpretations of these “(d , e, k )”

identities (S.)

Letd ∈Nand let 1≤i ≤k.

LetG_d,k,i(n)denote the number of partitionsπ ofnsuch that m_d(π)≤i−1 andm_dj(π) +m_dj+d(π)≤k −1 for anyj ∈N.

LetH_d,k,i(n)denote the number of partitions ofninto parts 6≡0,±di (mod 2d(k +1)).

ThenG_d,k_,i(n) =H_d,k,i(n)for all integersn.

This is a combinatorial interpretation of of the identity obtained by inserting the Bailey pair(α^(d,1,k_n ⁾(1,q), β_n^(d,1,k)(1,q))into (PBL) (along with associated systems ofq-difference equations).

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Combinatorial interpretations of these “(d , e, k )”

identities (S.)

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Combinatorial interpretations of these “(d , e, k )”

identities (S.)

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Combinatorial interpretations of these “(d , e, k )”

identities (S.)

This is a combinatorial interpretation of of the identity obtained by inserting the Bailey pair(α^(d,1,k)_n (1,q), β_n^(d,1,k)(1,q))into (PBL) (along with associated systems ofq-difference

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WHO

(outside the partitions andq-series community)

CARES?

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Connections to Lie algebras

In the 1980’s J. Lepowsky and R. Wilson showed that the principally specialized characters of standard modules for the odd levels ofA⁽¹⁾₁ are given by the The

Andrews–Gordon identity.

The two Rogers–Ramanujan identities occur at level 3. The even levels ofA⁽¹⁾₁ correspond to D. Bressoud’s even modulus analog of Andrews–Gordon.

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Connections to Lie algebras

The two Rogers–Ramanujan identities occur at level 3.

The even levels ofA⁽¹⁾₁ correspond to D. Bressoud’s even modulus analog of Andrews–Gordon.

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Connections to Lie algebras

The two Rogers–Ramanujan identities occur at level 3.

The even levels ofA⁽¹⁾ correspond to D. Bressoud’s even

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Capparelli’s identities (1988)

The Rogers–Ramanujan identities also occur at level 2 ofA⁽²⁾₂ .

Performing an analogous analysis of the level 3 modules ofA⁽²⁾₂ , S. Capparelli discovered:

The number of partitions ofninto parts≡ ±2,±3 (mod 12) equals the number of partitions(λ₁, λ₂, . . . , λ_l)ofnwhere

λ_i−λ_i+1≥2,

λ_i−λ_i+1=2 =⇒ λ_i ≡1 (mod 3), λi−λi+1=3 =⇒ λi ≡0 (mod 3)

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Capparelli’s identities (1988)

λ_i−λ_i+1≥2,

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Capparelli’s identities (1988)

λ_i−λ_i+1≥2,

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Analytic versions of Capparelli’s identity (S.)

1+ X

n,j,r≥0 (n,j,r)6=(0,0,0)

q³ⁿ²⁺⁹²^r²^+3j²+6nj+6nr+6rj−⁵₂r−j(1+q^2r+2j)(1−q^6r^+6j) (q³;q³)n(q³;q³)r(q³;q³)j(−1;q³)j+1(q³;q³)n+2r+2j

= 1

(q²,q³,q⁹,q¹⁰;q¹²)∞

X

n,j≥0

qⁿ²_n−j+1

3

(q)2n−j(q)_j = 1

(q²,q³,q⁹,q¹⁰;q¹²)∞

.

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Analytic versions of Capparelli’s identity (S.)

1+ X

n,j,r≥0 (n,j,r)6=(0,0,0)

q³ⁿ²⁺⁹²^r²^+3j²+6nj+6nr+6rj−⁵₂r−j(1+q^2r+2j)(1−q^6r^+6j) (q³;q³)n(q³;q³)r(q³;q³)j(−1;q³)j+1(q³;q³)n+2r+2j

= 1

(q²,q³,q⁹,q¹⁰;q¹²)∞

X

n,j≥0

qⁿ²_n−j+1

3

(q)2n−j(q)_j = 1

(q²,q³,q⁹,q¹⁰;q¹²)∞

.

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A

⁽²⁾₂

level 4 identities

In an analogous study of the level 4 modules ofA⁽²⁾₂ , D. Nandi (2014) conjectured three partition identities.

Proved by Motoki Takigiku and Shunsuke Tsuchioka (2019).

One of these identities is:

The number of partitions ofninto parts≡ ±2,±3,±4 (mod 14) equals the number of partitions(λ₁, λ₂, . . . , λ_l)ofnwhere

λ_i−λ_i+1≥2 λ_i−λ_i+2≥3

λ_i−λ_i+2=3 =⇒ λ_i 6=λ_i+1,

λ_i−λ_i+2=3 and 2-λ_i =⇒ λ_i+16=λ_i+2. λ_i−λ_i+2=4 and 2-λ_i =⇒ λ_i 6=λ_i+1, Consider the first differences

∆λ:= (λ₁−λ₂, λ₂−λ₃, . . . , λl−1−λ_l). None of the following subwords are permitted in∆λ:

(3,3,0),(3,2,3,0),(3,2,2,3,0), . . . ,(3,2,2,2,2, . . . ,2,3,0).

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A

⁽²⁾₂

level 4 identities

In an analogous study of the level 4 modules ofA⁽²⁾₂ , D. Nandi (2014) conjectured three partition identities. Proved by Motoki Takigiku and Shunsuke Tsuchioka (2019).

λ_i−λ_i+1≥2 λ_i−λ_i+2≥3

λ_i−λ_i+2=3 =⇒ λ_i 6=λ_i+1,

(3,3,0),(3,2,3,0),(3,2,2,3,0), . . . ,(3,2,2,2,2, . . . ,2,3,0).

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A

⁽²⁾₂

level 4 identities

λ_i−λ_i+1≥2 λ_i−λ_i+2≥3

λ_i−λ_i+2=3 =⇒ λ_i 6=λ_i+1,

(3,3,0),(3,2,3,0),(3,2,2,3,0), . . . ,(3,2,2,2,2, . . . ,2,3,0).

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A

⁽²⁾₂

level 4 identities

λ_i−λ_i+1≥2 λ_i−λ_i+2≥3

λ_i−λ_i+2=3 =⇒ λ_i 6=λ_i+1,

(3,3,0),(3,2,3,0),(3,2,2,3,0), . . . ,(3,2,2,2,2, . . . ,2,3,0).

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Shashank Kanade and Matthew Russell (2014)

Related to level 3 standard modules ofD₄⁽³⁾, Kandade and Russell conjectured several partition identities, including:

The number of partitions ofninto parts≡ ±1,±3 (mod 9) equals the number of partitionsλofnsuch that

λj−λj+2≥3,

λ_j−λ_j+1≤1 =⇒ 3|(λ_j+λ_j+1).

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Shashank Kanade and Matthew Russell (2014)

Related to level 3 standard modules ofD₄⁽³⁾, Kandade and Russell conjectured several partition identities, including:

The number of partitions ofninto parts≡ ±1,±3 (mod 9) equals the number of partitionsλofnsuch that

λj−λj+2≥3,

λ_j−λ_j+1≤1 =⇒ 3|(λ_j+λ_j+1).

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Kanade–Russell conjectures

Kanade and Russell have released a steady stream ofq-series and partition identity conjectures over the past six years.

Many have been proved by

Katherin Bringmann, Chris Jennings-Shaffer, and Karl Mahlburg;

Kagan Kur¸sungöz; Hjalmar Rosengren;

Kanade and Russell themselves.

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Kanade–Russell conjectures

Kanade and Russell have released a steady stream ofq-series and partition identity conjectures over the past six years. Many have been proved by

Katherin Bringmann, Chris Jennings-Shaffer, and Karl Mahlburg;

Kagan Kur¸sungöz;

Hjalmar Rosengren;

Kanade and Russell themselves.

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WHO ELSE

CARES?

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Polynomial RR identities

D₀(q) =D₁(q) =1

D_n(q) =D_n−1(q) +qⁿ⁻¹D_n−2ifn=2

Dn(q) =X

j=0

q^j² n−j

j

q

(MacMahon)

=X

j∈Z

(−1)^jq^j(5j+1)/2 n

b^n+5j+1₂ c

q

(Schur)

=X

k∈Z

q^k^(10k+1)τ0(n,5k;q)−q(5k+3)(2k+1)τ0(n,5k+3;q) (Andrews)

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Polynomial RR identities

D₀(q) =D₁(q) =1

Dn(q) =X

j=0

q^j² n−j

j

q

(MacMahon)

=X

j∈Z

(−1)^jq^j(5j+1)/2 n

b^n+5j+1₂ c

q

(Schur)

=X

k∈Z

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Polynomial RR identities

D₀(q) =D₁(q) =1

Dn(q) =X

j=0

q^j² n−j

j

q

(MacMahon)

=X

j∈Z

(−1)^jq^j(5j+1)/2 n

b^n+5j+1₂ c

q

(Schur)

=X

k∈Z

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Polynomial RR identities

D₀(q) =D₁(q) =1

Dn(q) =X

j=0

q^j² n−j

j

q

(MacMahon)

=X

j∈Z

(−1)^jq^j(5j+1)/2 n

b^n+5j+1₂ c

q

(Schur)

=X

k∈Z

q^k^(10k+1)τ0(n,5k;q)−q(5k+3)(2k+1)τ0(n,5k+3;q)

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Polynomial RR identities

We can prove these polynomial identities via recurrences, and then the original series–infinite product identity follows via asymptotics ofq-bi/trinomial coëfficients, and the triple product identity.

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q-binomial and q-trinomial coëfficients

A B

q

:= (q)_A(q)⁻¹_B (q)⁻¹_A−Bif 05B5A; 0 o/w

T₀(L,A;q) :=

L

X

r=0

(−1)^r L

r

q²

2L−2r L−A−r

q

T₁(L,A;q) :=

L

X

r=0

(−q)^r L

r

q²

2L−2r L−A−r

q

τ0(L,A;q) :=

L

X

r=0

(−1)^rq^Lr−(^r2) L r

q

2L−2r L−A−r

q