new mock theta function identitiesfishel/talkstopost/garvan.pdf · 2018. 1. 12. · a mock theta...

Outline ABSTRACT RAMANUJAN’s LAST LETTER INDEFINITE THETA SERIES MOCK THETA ORDER 7 RELATIONS BAILEY TRANSFORM APPLICATIONS REFERENCES

New Mock ThetaFunction Identities

F.G. Garvanurl: qseries.org/fgarvan

University of Florida

JMM2018, San Diego — Saturday, January 13, 2018

F.G. Garvan url: qseries.org/fgarvan New Mock Theta Function Identities

1 ABSTRACT

2 RAMANUJAN’s LAST LETTER

3 INDEFINITE THETA SERIESMISSING FIFTH ORDER FUNCTIONSSEVENTH ORDER FUNCTIONS

4 MOCK THETA ORDER 7 RELATIONS

5 BAILEY TRANSFORMBAILEY PAIRSNEW CONJUGATE BAILEY PAIR

6 APPLICATIONSHECKE DOUBLE SUM FOR χ1(q)HECKE DOUBLE SUM FOR F2(q)

7 REFERENCES

ABSTRACT

In his last letter to Hardy, Ramanujan defined ten mock theta functions of order 5 and three of order 7. He stated

that the three mock theta functions of order 7 are not related. We give simple proofs of new Hecke double sum

identities for two of the order 5 functions and all three of the order 7 functions. We find that the coefficients of

Ramanujan’s three mock theta functions of order 7 are surprisingly related.

RAMANUJAN (1887 – 1920)

RAMANUJAN’s LAST LETTER

RAMANUJAN’S DEFINITION OF A MOCK THETAFUNCTIONA mock theta function is a function f of a complex variable q,defined by a q-series of a particular type (Eulerian form), whichconverges for |q| < 1 and satisfies the following conditions:

infinitely many roots of unity are exponential singularities;

for every root of unity ξ, there is a theta function ϑξ(q) suchthe difference f (q)− ϑξ(q) is bounded as q → ξ radially; and

f is not the sum of two functions, one of which is a thetafunction and the other a function that is bounded radiallytoward all roots of unity.

MODERN DEFINITION OF A MOCK THETA FUNCTION(Zwegers (2002), Zagier (2010))A mock theta function is a q-series H(q) =

∑∞n=0 anq

n such thatthere is a rational number λ ∈ Q and a unary theta functiong(z) =

∑n∈Z+α nq

κn2such that the function

h(z) = qλH(q) + g∗(z)

is a nonholomorphic modular form of weight 1/2, where

g∗(z) =∑

n∈Z+α

sgn(n)β(4κn2y)q−κn2,

q = exp(2πiz), z = x + iy and β is an incomplete gamma function.Rhoades (2013) Griffin, Ono and Rolen (2013)

∑∞n=0 anq

∑n∈Z+α nq

h(z) = qλH(q) + g∗(z)

g∗(z) =∑

n∈Z+α

∑∞n=0 anq

∑n∈Z+α nq

h(z) = qλH(q) + g∗(z)

g∗(z) =∑

n∈Z+α

∑∞n=0 anq

∑n∈Z+α nq

h(z) = qλH(q) + g∗(z)

g∗(z) =∑

n∈Z+α

∑∞n=0 anq

∑n∈Z+α nq

h(z) = qλH(q) + g∗(z)

g∗(z) =∑

n∈Z+α

EXAMPLEZwegers (2001)

f (q) = 1 +∞∑n=1

(1 + q)2(1 + q2)2 · · · (1 + qn)2

Thenh(z) = q−1/24f (q) + g∗(z)

transforms like a 1/2 modular form on Γ(2),

g∗(z) =∑

n≡1 (mod 6)

sgn(n)β(n2y/6)q−n2/24

f (q) = 1 +∞∑n=1

(1 + q)2(1 + q2)2 · · · (1 + qn)2

Thenh(z) = q−1/24f (q) + g∗(z)

g∗(z) =∑

n≡1 (mod 6)

f (q) = 1 +∞∑n=1

(1 + q)2(1 + q2)2 · · · (1 + qn)2

Thenh(z) = q−1/24f (q) + g∗(z)

g∗(z) =∑

n≡1 (mod 6)

f (q) = 1 +∞∑n=1

(1 + q)2(1 + q2)2 · · · (1 + qn)2

Thenh(z) = q−1/24f (q) + g∗(z)

g∗(z) =∑

n≡1 (mod 6)

Outline ABSTRACT RAMANUJAN’s LAST LETTER INDEFINITE THETA SERIES MOCK THETA ORDER 7 RELATIONS BAILEY TRANSFORM APPLICATIONS REFERENCESMISSING FIFTH ORDER FUNCTIONS SEVENTH ORDER FUNCTIONS

INDEFINITE THETA SERIES

Andrews (1986)Indefinite theta series for all but two of the 5th order functions.All three of the 7th order functionsEXAMPLEDefine

f0(q) = 1 +∞∑n=1

(1 + q)(1 + q2) · · · (1 + qn)

f0(q) =∞∏n=1

(1− qn)

∞∑n=0

n∑j=−n

(−1)jqn(5n+1)/2−j2(1− q4n+2)

f0(q) = 1 +∞∑n=1

(1 + q)(1 + q2) · · · (1 + qn)

f0(q) =∞∏n=1

(1− qn)

∞∑n=0

n∑j=−n

(−1)jqn(5n+1)/2−j2(1− q4n+2)

f0(q) = 1 +∞∑n=1

(1 + q)(1 + q2) · · · (1 + qn)

f0(q) =∞∏n=1

(1− qn)

∞∑n=0

n∑j=−n

(−1)jqn(5n+1)/2−j2(1− q4n+2)

f0(q) = 1 +∞∑n=1

(1 + q)(1 + q2) · · · (1 + qn)

f0(q) =∞∏n=1

(1− qn)

∞∑n=0

n∑j=−n

(−1)jqn(5n+1)/2−j2(1− q4n+2)

f0(q) = 1 +∞∑n=1

(1 + q)(1 + q2) · · · (1 + qn)

f0(q) =∞∏n=1

(1− qn)

∞∑n=0

n∑j=−n

(−1)jqn(5n+1)/2−j2(1− q4n+2)

NOTATION

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

(a)∞ = (a; q)∞ =∞∏n=1

(1− aqn−1)

NOTATION

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

(a)∞ = (a; q)∞ =∞∏n=1

(1− aqn−1)

MISSING FIFTH ORDER FUNCTIONS

χ0(q) =∞∑n=0

(qn+1; q)n=∞∑n=0

qn (q)n(q)2n

1 + q + q2 + 2 q3 + q4 + 3 q5 + 2 q6 + 3 q7 + · · · ,

χ1(q) =∞∑n=0

(qn+1; q)n+1=∞∑n=0

qn (q)n(q)2n+1

= 1 + 2 q + 2 q2 + 3 q3 + 3 q4 + 4 q5 + 4 q6 + 6 q7 + · · · .

Zwegers (2009) triple sumsZagier (2007–2009) double sums

χ0(q) =∞∑n=0

(qn+1; q)n=∞∑n=0

qn (q)n(q)2n

1 + q + q2 + 2 q3 + q4 + 3 q5 + 2 q6 + 3 q7 + · · · ,

χ1(q) =∞∑n=0

(qn+1; q)n+1=∞∑n=0

qn (q)n(q)2n+1

= 1 + 2 q + 2 q2 + 3 q3 + 3 q4 + 4 q5 + 4 q6 + 6 q7 + · · · .

χ0(q) =∞∑n=0

(qn+1; q)n=∞∑n=0

qn (q)n(q)2n

1 + q + q2 + 2 q3 + q4 + 3 q5 + 2 q6 + 3 q7 + · · · ,

χ1(q) =∞∑n=0

(qn+1; q)n+1=∞∑n=0

qn (q)n(q)2n+1

= 1 + 2 q + 2 q2 + 3 q3 + 3 q4 + 4 q5 + 4 q6 + 6 q7 + · · · .

χ0(q) =∞∑n=0

(qn+1; q)n=∞∑n=0

qn (q)n(q)2n

1 + q + q2 + 2 q3 + q4 + 3 q5 + 2 q6 + 3 q7 + · · · ,

χ1(q) =∞∑n=0

(qn+1; q)n+1=∞∑n=0

qn (q)n(q)2n+1

= 1 + 2 q + 2 q2 + 3 q3 + 3 q4 + 4 q5 + 4 q6 + 6 q7 + · · · .

ZAGIER’S IDENTITIES (2007-2009)

(q)∞(2− χ0(q))

5|b|<|a|a+b≡2 (mod 4)a≡2 (mod 5)

sgn(a)

a2 − b2

(a2−5b2)− 130

(q)∞χ1(q))

5|b|<|a|a+b≡2 (mod 4)a≡4 (mod 5)

sgn(a)

a2 − b2

(a2−5b2)− 1930

ZAGIER’S IDENTITIES (2007-2009)

(q)∞(2− χ0(q))

5|b|<|a|a+b≡2 (mod 4)a≡2 (mod 5)

sgn(a)

a2 − b2

(a2−5b2)− 130

(q)∞χ1(q))

5|b|<|a|a+b≡2 (mod 4)a≡4 (mod 5)

sgn(a)

a2 − b2

(a2−5b2)− 1930

THEOREM [G. (2016)]

(q)∞(2− χ0(q))

3|b|<5|a|a≡1 (mod 6)

b≡1,11 (mod 30)

sgn(b)

)χ60(b)q

(5a2−b2)− 130

(q)∞χ1(q))

= i∑

3|b|<5|a|a≡b≡1 (mod 6)b≡±2 (mod 5)

sgn(b)

)χ60(b)q

(5a2−b2)− 1930

THEOREM [G. (2016)]

(q)∞(2− χ0(q))

3|b|<5|a|a≡1 (mod 6)

b≡1,11 (mod 30)

sgn(b)

)χ60(b)q

(5a2−b2)− 130

(q)∞χ1(q))

= i∑

3|b|<5|a|a≡b≡1 (mod 6)b≡±2 (mod 5)

sgn(b)

)χ60(b)q

(5a2−b2)− 1930

SEVENTH ORDER FUNCTIONS

F0(q) =∞∑n=0

(qn+1; q)n

= 1 + q + q3 + q4 + q5 + 2q7 + q8 + 2q9 + · · · ,

F1(q) =∞∑n=0

(qn; q)n

= q + q2 + q3 + 2q4 + q5 + 2q6 + 2q7 + 2q8 + 3q9 + · · · ,

F2(q) =∞∑n=0

(qn+1; q)n+1

= 1 + q + 2q2 + q3 + 2q4 + 2q5 + 3q6 + 2q7 + 3q8 + 3q9 + · · · .

ANDREWS’S IDENTITIES (1986)

F0(q) =1

(q)∞

∞∑n=0

n∑j=−n

q7n2+n−j2(1− q12n+6)

−2q∞∑n=0

n∑j=0

q7n2+8n−j2−j(1− q12n+12)

F1(q) = · · ·F2(q) = · · ·

MORE ZAGIER IDENTITIES (2007-2009)

(q7; q7)∞∑|b|<|a|, ab>0a≡4 (mod 7)b≡3 (mod 7)

sgn(a) (2ε6(b)− ε2(a)ε3(b)− ε3(a)ε2(b)) q1

42(ab)− 2

F1(q) = · · ·F2(q) = · · ·

where εN(a) = 1 if a ≡ 0 (mod N) and 0 otherwise.

MORE ZAGIER IDENTITIES (2007-2009)

(q; q)3∞∑

2|b|/9<ab≡1 (mod 7)

)(a sgn(b)− 3n

18a2− 1

168b2− 5

F1(q) = · · ·F2(q) = · · ·

THEOREM [G. (2016)] Define

G∗(r1, r2, s; q)

3|b|<7|a|a≡1 (mod 6)

b≡r1,r2 (mod 42)

sgn(b)

(7a2−b2)+s

F0(q) = − 1

(q)∞G∗(1, 13,−1/28)

F1(q) = − 1

(q)∞G∗(5, 19, 3/28)

F2(q) =1

(q)∞G∗(11, 17,−9/28)

G∗(r1, r2, s; q)

3|b|<7|a|a≡1 (mod 6)

b≡r1,r2 (mod 42)

sgn(b)

(7a2−b2)+s

F0(q) = − 1

(q)∞G∗(1, 13,−1/28)

F1(q) = − 1

(q)∞G∗(5, 19, 3/28)

F2(q) =1

(q)∞G∗(11, 17,−9/28)

G∗(r1, r2, s; q)

3|b|<7|a|a≡1 (mod 6)

b≡r1,r2 (mod 42)

sgn(b)

(7a2−b2)+s

F0(q) = − 1

(q)∞G∗(1, 13,−1/28)

F1(q) = − 1

(q)∞G∗(5, 19, 3/28)

F2(q) =1

(q)∞G∗(11, 17,−9/28)

G∗(r1, r2, s; q)

3|b|<7|a|a≡1 (mod 6)

b≡r1,r2 (mod 42)

sgn(b)

(7a2−b2)+s

F0(q) = − 1

(q)∞G∗(1, 13,−1/28)

F1(q) = − 1

(q)∞G∗(5, 19, 3/28)

F2(q) =1

(q)∞G∗(11, 17,−9/28)

MOCK THETA ORDER 7 RELATIONS

For j = 0, 1, 2 define cj(n) by

∞∑n=0

cj(n)qn = (q)∞Fj(q)

THEOREM [G. (2016)]

If p ≡ ±5 (mod 28) then

c0(p2n + 128 (9p2 − 1)) = ±c2(n)

c1(p2n + 128 (p2 + 3)) = ±c0(n)

c2(p2n + 128 (25p2 − 9)) = ∓c1(n + 1)

MOCK THETA ORDER 7 RELATIONS

For j = 0, 1, 2 define cj(n) by

∞∑n=0

cj(n)qn = (q)∞Fj(q)

THEOREM [G. (2016)]

c0(p2n + 128 (9p2 − 1)) = ±c2(n)

c1(p2n + 128 (p2 + 3)) = ±c0(n)

c2(p2n + 128 (25p2 − 9)) = ∓c1(n + 1)

c0(p2n + 128 (25p2 − 1)) = ∓c1(n + 1)

c1(p2n + 128 (9p2 + 3)) = ±c2(n)

c2(p2n + 128 (p2 − 9)) = ∓c0(n)

c0(p2n + 128 (p2 − 1)) = ∓c0(n)

c1(p2n + 128 (25p2 + 3)) = ∓c1(n + 1)

c2(p2n + 128 (9p2 − 9)) = ∓c2(n)

c0(p2n + 128 (25p2 − 1)) = ∓c1(n + 1)

c1(p2n + 128 (9p2 + 3)) = ±c2(n)

c2(p2n + 128 (p2 − 9)) = ∓c0(n)

c0(p2n + 128 (p2 − 1)) = ∓c0(n)

c1(p2n + 128 (25p2 + 3)) = ∓c1(n + 1)

c2(p2n + 128 (9p2 − 9)) = ∓c2(n)

Outline ABSTRACT RAMANUJAN’s LAST LETTER INDEFINITE THETA SERIES MOCK THETA ORDER 7 RELATIONS BAILEY TRANSFORM APPLICATIONS REFERENCESBAILEY PAIRS NEW CONJUGATE BAILEY PAIR

BAILEY TRANSFORM

βn =n∑

αrun−rvn+r

γn =∞∑r=n

δrur−nvr+n

then∞∑n=0

anγn =∞∑n=0

βnδn

(assuming convergence)

BAILEY TRANSFORM

βn =n∑

αrun−rvn+r

γn =∞∑r=n

δrur−nvr+n

then∞∑n=0

anγn =∞∑n=0

βnδn

BAILEY TRANSFORM

βn =n∑

αrun−rvn+r

γn =∞∑r=n

δrur−nvr+n

then∞∑n=0

anγn =∞∑n=0

βnδn

BAILEY PAIRS

A pair of sequences (αn, βn)n≥0 is called a Bailey pair relative to(a, q) if

βn =n∑

(q; q)n−r (aq; q)n+r

A pair of sequences (γn, δn)n≥0 is called a conjugate Bailey pairrelative to (a, q) if

γn =∞∑r=n

δr(q; q)r−n(aq; q)r+n

BAILEY PAIRS

A pair of sequences (αn, βn)n≥0 is called a Bailey pair relative to(a, q) if

βn =n∑

(q; q)n−r (aq; q)n+r

A pair of sequences (γn, δn)n≥0 is called a conjugate Bailey pairrelative to (a, q) if

γn =∞∑r=n

δr(q; q)r−n(aq; q)r+n

NEW CONJUGATE BAILEY PAIR

relative to (q, q)

δn =qn(q; q)n(q; q)∞

(1− q)

γn =∞∑

(−1)j+n+1qj(3j−1)/2−3n(n+1)/2−1(1 + qj)

We need ANDREWS-SUBBARAO-FINE-ROGERS identity:

∞∑n=1

(xq)n−1xnqn =

∞∑n=1

(−1)n−1x3n−2qn(3n−1)/2(1 + xqn)

FRANKLIN INVOLUTION

relative to (q, q)

(1− q)

γn =∞∑

(−1)j+n+1qj(3j−1)/2−3n(n+1)/2−1(1 + qj)

∞∑n=1

(xq)n−1xnqn =

∞∑n=1

(−1)n−1x3n−2qn(3n−1)/2(1 + xqn)

FRANKLIN INVOLUTION

relative to (q, q)

(1− q)

γn =∞∑

(−1)j+n+1qj(3j−1)/2−3n(n+1)/2−1(1 + qj)

∞∑n=1

(xq)n−1xnqn =

∞∑n=1

(−1)n−1x3n−2qn(3n−1)/2(1 + xqn)

FRANKLIN INVOLUTION

γn =∞∑r=n

δr(q)r−n(q2; q)r+n

= (q)∞

∞∑r=n

(q)rqr

(q)r−n(q; q)r+n+1

= (q)∞

∞∑r=0

(q)r+nqr+n

(q)r (q; q)r+2n+1

= (q)∞qn(q)n

(q)2n+1

∞∑r=0

(qn+1; q)rqr

(q)r (q2n+2; q)r

γn =∞∑r=n

= (q)∞

∞∑r=n

(q)rqr

(q)r−n(q; q)r+n+1

= (q)∞

∞∑r=0

(q)r+nqr+n

(q)r (q; q)r+2n+1

= (q)∞qn(q)n

(q)2n+1

∞∑r=0

(qn+1; q)rqr

(q)r (q2n+2; q)r

γn =∞∑r=n

= (q)∞

∞∑r=n

(q)rqr

(q)r−n(q; q)r+n+1

= (q)∞

∞∑r=0

(q)r+nqr+n

(q)r (q; q)r+2n+1

= (q)∞qn(q)n

(q)2n+1

∞∑r=0

(qn+1; q)rqr

(q)r (q2n+2; q)r

γn =∞∑r=n

= (q)∞

∞∑r=n

(q)rqr

(q)r−n(q; q)r+n+1

= (q)∞

∞∑r=0

(q)r+nqr+n

(q)r (q; q)r+2n+1

= (q)∞qn(q)n

(q)2n+1

∞∑r=0

(qn+1; q)rqr

(q)r (q2n+2; q)r

GASPER AND RAHMAN

q-HYPERGEOMETRIC NOTATION

(a, b; q, z

∞∑n=0

(a; q)n(b; q)nzn

(c ; q)n(q; q)n

= 1 +(1− a)(1− b)z

(1− c)(1− q)+

(1− a)(1− aq)(1− b)(1− bq)z2

(1− c)(1− cq)(1− q)(1− q2)+ · · ·

PROOF CONTINUED

γn = (q)∞qn(q)n

(q)2n+12φ1

(0, qn+1; q, q

)= (q)∞qn

(q)n(q)2n+1

(qn+1; q)∞(q2n+2; q)∞(q)∞

(0, qn+1; q, qn+1

)(by Heine’s transformation)

= qn∞∑j=0

(qn+1; q)j(qn+1

)j= · · ·

PROOF CONTINUED

γn = (q)∞qn(q)n

(q)2n+12φ1

(0, qn+1; q, q

)= (q)∞qn

(q)n(q)2n+1

(qn+1; q)∞(q2n+2; q)∞(q)∞

(0, qn+1; q, qn+1

= qn∞∑j=0

(qn+1; q)j(qn+1

)j= · · ·

PROOF CONTINUED

γn = (q)∞qn(q)n

(q)2n+12φ1

(0, qn+1; q, q

)= (q)∞qn

(q)n(q)2n+1

(qn+1; q)∞(q2n+2; q)∞(q)∞

(0, qn+1; q, qn+1

= qn∞∑j=0

(qn+1; q)j(qn+1

)j= · · ·

PROOF CONTINUED

γn = (q)∞qn(q)n

(q)2n+12φ1

(0, qn+1; q, q

)= (q)∞qn

(q)n(q)2n+1

(qn+1; q)∞(q2n+2; q)∞(q)∞

(0, qn+1; q, qn+1

= qn∞∑j=0

(qn+1; q)j(qn+1

)j= · · ·

Outline ABSTRACT RAMANUJAN’s LAST LETTER INDEFINITE THETA SERIES MOCK THETA ORDER 7 RELATIONS BAILEY TRANSFORM APPLICATIONS REFERENCESHECKE DOUBLE SUM FOR χ1(q) HECKE DOUBLE SUM FOR F2(q)

APPLICATIONS

HECKE DOUBLE SUM FOR χ1(q)

χ1(q) =∞∑n=0

(qn+1; q)n+1

=∞∑n=0

(q)nqn

(q)2n+1

=∞∑n=0

qn(q)n(1− q)

(q2; q)2n

χ1(q) =∞∑n=0

(qn+1; q)n+1

=∞∑n=0

(q)nqn

(q)2n+1

=∞∑n=0

qn(q)n(1− q)

(q2; q)2n

χ1(q) =∞∑n=0

(qn+1; q)n+1

=∞∑n=0

(q)nqn

(q)2n+1

=∞∑n=0

qn(q)n(1− q)

(q2; q)2n

SLATER A(2) BAILEY PAIR relative to (q, q)

βn = 1(q2;q)2n

α3m−1 = q6m2−m

α3m = q6m2+m

α3m+1 = −q6m2+5m+1 − q6m2+7m+2

(q)∞χ1(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(15m−7)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(15m+17)/2−3(1 + qj)

+qj(3j−1)/2−m(15m+13)/2−2(1 + qj)}

(q)∞χ1(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(15m−7)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(15m+17)/2−3(1 + qj)

+qj(3j−1)/2−m(15m+13)/2−2(1 + qj)}

(q)∞χ1(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(15m−7)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(15m+17)/2−3(1 + qj)

+qj(3j−1)/2−m(15m+13)/2−2(1 + qj)}

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+13)/2−2(1 + qj)

j(3j−1)/2−m(15m+7)/2−1 =1

(5(6j − 1)2 − (30m + 7)2

)−19

j(3j−1)/2−m(15m+13)/2−2 =1

(5(6j − 1)2 − (30m + 13)2

)−19

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+13)/2−2(1 + qj)

j(3j−1)/2−m(15m+7)/2−1 =1

(5(6j − 1)2 − (30m + 7)2

)−19

j(3j−1)/2−m(15m+13)/2−2 =1

(5(6j − 1)2 − (30m + 13)2

)−19

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+7)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(15m+13)/2−2(1 + qj)

j(3j−1)/2−m(15m+7)/2−1 =1

(5(6j − 1)2 − (30m + 7)2

)−19

j(3j−1)/2−m(15m+13)/2−2 =1

(5(6j − 1)2 − (30m + 13)2

)−19

HECKE DOUBLE SUM FOR F2(q)

F2(q) =∞∑n=0

(qn+1; q)n+1

=∞∑n=0

qn(q)n(1− q)

(q2; q)2n

HECKE DOUBLE SUM FOR F2(q)

F2(q) =∞∑n=0

(qn+1; q)n+1

=∞∑n=0

qn(q)n(1− q)

(q2; q)2n

SLATER A(6) BAILEY PAIR relative to (q, q)

βn = qn2

(q2;q)2nα3m−1 = q3m2+m

α3m = q3m2−m

α3m+1 = −q3m2+m − q3m2+5m+2

(q)∞F2(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(21m−11)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(21m+25)/2−4(1 + qj)

+qj(3j−1)/2−m(21m+17)/2−2(1 + qj)}

(q)∞F2(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(21m−11)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(21m+25)/2−4(1 + qj)

+qj(3j−1)/2−m(21m+17)/2−2(1 + qj)}

(q)∞F2(q) =∞∑n=0

δnβn

=∞∑

αmγm

=∞∑

∞∑j=3m

(−1)m+jqj(3j−1)/2−m(21m−11)/2−1(1 + qj)

+∞∑

∞∑j=3m+1

(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑

∞∑j=3m+2

(−1)m+j+1{qj(3j−1)/2−m(21m+25)/2−4(1 + qj)

+qj(3j−1)/2−m(21m+17)/2−2(1 + qj)}

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+17)/2−2(1 + qj)

j(3j−1)/2−m(21m+11)/2−1 =1

(7(6j − 1)2 − (42m + 11)2

)− 9

j(3j−1)/2−m(21m+17)/2−2 =1

(7(6j − 1)2 − (42m + 17)2

)− 9

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+17)/2−2(1 + qj)

j(3j−1)/2−m(21m+11)/2−1 =1

(7(6j − 1)2 − (42m + 11)2

)− 9

j(3j−1)/2−m(21m+17)/2−2 =1

(7(6j − 1)2 − (42m + 17)2

)− 9

=∞∑j=1

∑−j≤3m≤j−1

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+11)/2−1(1 + qj)

+∞∑j=1

∑−j−1≤3m≤j−2

sgn(m)(−1)m+j+1qj(3j−1)/2−m(21m+17)/2−2(1 + qj)

j(3j−1)/2−m(21m+11)/2−1 =1

(7(6j − 1)2 − (42m + 11)2

)− 9

j(3j−1)/2−m(21m+17)/2−2 =1

(7(6j − 1)2 − (42m + 17)2

)− 9

HAPPY BIRTHDAY DENNIS

THANK YOU

REFERENCES

G. E. Andrews, The fifth and seventh order mock thetafunctions, Trans. Amer. Math. Soc. 293 (1986), 113–134.G. Gasper and M. Rahman, Basic Hypergeometric Series,Encycl. Math. Appl., Cambridge Univ. Press, Cambridge,2004.L. J. Slater, A new proof of Rogers’s transformations ofinfinite series, Proc. London Math. Soc. (2) 53 (1951),460–475.D. Zagier, Ramanujan’s mock theta functions and theirapplications (after Zwegers and Ono-Bringmann), Asterisque326 (2009), Exp. No. 986, vii–viii, 143–164 (2010).Seminaire Bourbaki. Vol. 2007/2008.S. Zwegers, On two fifth order mock theta functions,Ramanujan J. 20 (2009), 207–214.S. P. Zwegers, “Mock Theta Functions,” Ph.D. thesis,Universiteit Utrecht, 2002, 96 pp.

new mock theta function identitiesfishel/talkstopost/garvan.pdf · 2018. 1. 12. · a mock theta...

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