FOUR IDENTITIES RELATED TO THIRD ORDER MOCK THETA FUNCTIONS IN RAMANUJAN S LOST NOTEBOOK HAMZA YESILYURT

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1 FOUR IDENTITIES RELATED TO THIRD ORDER MOCK THETA FUNCTIONS IN RAMANUJAN S LOST NOTEBOOK HAMZA YESILYURT Abstract. We prove, for the first time, a series of four related identities from Ramanujan s lost notebook. The identities are connected with third order mock theta functions.. Introduction In his last letter to Hardy, Ramanujan introduced mock theta functions [9, pp. 27-3]. Included in this letter were the four third order mock theta functions q f(q n2 ( + q 2 ( + q 2 2 ( + q n, (. 2 φ(q ψ(q χ(q They satisfy the equations n0 n0 n n0 where we used the standard notation q n2 ( + q 2 ( + q 4 ( + q 2n, (.2 q n2 ( q( q 3 ( q 2n, (.3 q n2 ( q + q 2 ( q 2 + q 4 ( q n + q 2n. (.4 2 φ( q f(q f(q + 4 ψ( q ( q; q n ( n q n2, (.5 4 χ(q f(q 3(q; q ( n q 3n2 2, (.6 n (a; q n (a n : ( a( aq... ( aq n, (a; q ( aq n, q <. n0 G. N. Watson [0] proved (.5 and (.6. G. E. Andrews [] also gave certain generalizations of (.5 and (.6. Third order mock theta functions are related to the rank of a partition defined

2 2 Mock Theta Function Identities by F. J. Dyson [5] as the largest part minus the number of parts. Let us define N(m, n as the number of partitions of n with rank m. The generating function for N(m, n is given by N(m, nq n t m q n2, q <, q < t < /q. (.7 (tq n (t q n m n0 n0 The third order mock theta functions defined by (. through (.4 can be expressed in terms of this generating function. Third order mock theta functions and their applications to the rank are detailed by N. J. Fine [7]. A comprehensive literature survey on mock theta functions is given by G. E. Andrews [2]. We prove, for the first time, a series of four related identities from Ramanujan s lost notebook. These identities are defined and their connections to (.5 and (.6 are given in 3. Proofs of these identities are provided in 4 through 7. In addition, we will show in 8 that one of the identities can be used to prove the following identity (q 2 (t (t q n ( n qn(n+/2 tq n. (.8 Identity (.8 was proved by R. J. Evans [6, eq. (3.] following a different approach. Equality (.8 is also given in a different form by Ramanujan on page 59 of the lost notebook [9]. Partition theory implications of the product (q (tq (t q are discussed in [8]. 2. Definitions and Preliminary Results We first recall Ramanujan s definitions for a general theta function and some of its important special cases. Set f(a, b : a n(n+/2 b n(n /2, ab <. (2. n Basic properties satisfied by f(a, b include [4, p. 34, Entry 8] and if n is an integer, If n, (2.5 becomes f(a, b f(b, a, (2.2 f(, a 2f(a, a 3, (2.3 f(, a 0, (2.4 f(a, b a n(n+/2 b n(n /2 f(a(ab n, b(ab n. (2.5 Two other formulas satisfied by f(a, b are [4, p. 46, Entry 30] f(a, b af(a, a 2 b. (2.6 f(a, b + f( a, b 2f(a 3 b, ab 3, (2.7 f(a, b f( a, b 2af(ba, ab a 4 b 4. (2.8

3 Mock Theta Function Identities 3 The function f(a, b satisfies the well-known Jacobi triple product identity [4, p. 35, Entry 9] f(a, b ( a; ab ( b; ab (ab; ab. (2.9 Some important special cases of (2. and (2.9 are ϕ(q : f(q, q q n2 ( q; q 2 2 (q 2 ; q 2, (2.0 ψ(q : f(q, q 3 n q n(n+/2 ( q; q 2 (q; q, (2. n0 f( q : f( q, q 2 n ( n q n(3n /2 (q; q. (2.2 By using (2.0 and (2., and elementary product manipulations, we find that Also, after Ramanujan define ψ( q ϕ( q (2.3 (q; q (q2 ; q 2, (q 2 ; q 4 ( q; q 2 (2.4 (q; q. ( q; q (2.5 χ(q : ( q; q 2. (2.6 Other basic properties of the functions ϕ, ψ, f and χ are [4, p. 39, Entry 24] f(q f( q ψ(q ψ( q χ(q χ( q ϕ(q ϕ( q, (2.7 χ(q Combining (2.8 and (2.9, we obtain f(q f( q 2 ϕ(q 3 ψ( q ϕ(q f(q f( q2 ψ( q, (2.8 f 3 ( q 2 ϕ( qψ 2 (q. (2.9 We will frequently use Euler s identity For any real number a, let f a (q : n0 f 3 ( q ψ(qϕ 2 ( q. (2.20 ( q; q (q; q 2. (2.2 q n2 ( + aq + q 2 ( + aq n + q 2n, (2.22

4 4 Mock Theta Function Identities where q <. For q <, q < t < q, let q n2 G(t, q :. (2.23 (tq n (t q n n0 We need Euler s famous generating function for partitions, G(, q (q; q. (2.24 For a proof of (2.24 see [7, p. 3, eq. (2.3]. We need variations of two representations for G(t, q due to Fine [7]. Lemma 2.. For t <, G(t, q ( t n0 t + t n0 t n (t q n (2.25 n0 Proof. Following Fine [7, pp., eq. (.], we define (aq n F (a, b; t : t n. (bq n In this notation, Lemma 2. can be written as (tq n (t q n. (2.26 G(t, q ( tf (0, t ; t (2.27 t + t F (0, t ; tq. (2.28 Equation (2.27 is equation (2.3 on page 3 of [7] with b replaced by t, and (2.28 readily follows from equation (2.4 on page 2 of [7]. Observe that (2.26 is valid in the region q < t < q. Also as noted by Fine [7, p. 5, eq. (25.6], G(t, q satisfies a third order q difference equation. We sketch a proof here since it is stated without a proof in [7]. Lemma 2.2. For q < and q < t < /q, G(t,q satisfies the q-difference equation Proof. Let qt3 G(tq, q + tq t G(t, q qt2. (2.29 M(t, q : n0 (tq n (t q n, (2.30 so that by (2.26, G(t, q t + t M(t, q. (2.3

5 Using the definition (2.30 and algebraic manipulation, we obtain Mock Theta Function Identities 5 (tq n (tq n ( t q n+ M(t, q (t q n0 n (t q n0 n+ (tq n t (tq 2 n q (t q n0 n+ (t q n0 n+ (tq n t q( t (tq 2 n (t q n0 n+ (t n0 n+2 (tq n+ tq (t q n0 n+ (tq 2 2 t q( t n0 ( M(t, q ( t M(tq, q tq t 3 q 3 (tq 2 n+2 (t n+2 tq2. (2.32 t Now, Lemma 2.2 follows from (2.32 together with (2.3 after rearrangement. For convenience, define Lemma 2.2 then takes the following form V (t, q : G(t, q. (2.33 t V (tq, q + qt 3 V (t, q qt 2. (2.34 Observe that V (t, q tv (t, q. (2.35 The basic property (2.35 will be used many times in the sequel without comment. The partial fraction decomposition of V (t, q is given by [8, eq. (7.0] V (t, q + t (q n ( n q3n(n+/2 tq n. (2.36 We will need the following lemma due to A. O. L. Atkin and P. Swinnerton-Dyer [3]. Lemma 2.3. Let q, 0 < q <, be fixed. Suppose that ϑ(z is an analytic function of z, except for possibly a finite number of poles, in every region, 0 < z z z 2. If ϑ(zq Az k ϑ(z for some integer k (positive, zero, or negative and some constant A, then either ϑ(z has k more poles than zeros in the region q < z, or ϑ(z vanishes identically.

6 6 Mock Theta Function Identities 3. Four Identities of Ramanujan We now offer the four identities from Ramanujan s lost notebook that we plan to prove. Entry 3.. [9, p. 2, no. 3] Suppose that a and b are real with a 2 + b 2 4. Then, if f a (q is defined by (2.22, b a f a ( q+ b + a + 2 f a ( q b 4 2 f b(q (q4 ; q 4 ( q; q 2 n bq n + q 2n. (3. + (a 2 b 2 2q 4n + q8n If we take a 0 and b 2, then, by using (2.5 and elementary product manipulations, we see that (3. reduces to (.5 in the notation of (2.0 as follows: 2 φ( q f(q ( q ϕ( q. Entry 3.2. [9, p. 2, no. 4] If a and b are real with a 2 + ab + b 2 3, then (a + f a (q+(b + f b (q (a + b f a+b (q 3 (q3 ; q 3 2. (3.2 (q; q + ab(a + bq 3n + q6n In (3.2, take a b and use (2.5; then one obtains (.6 in the notation of (2.0 as n 4 χ(q f(q 3(q ϕ 2 ( q 3. We changed the notation that Ramanujan used in the left hand side of the next entry to avoid confusion. Also note that the series on the right side below is f 3 (q in the notation of (2.22. Entry 3.3. [9, p. 7, no. 5] With f a (q defined by (2.22, + 3 f ( q f ( q 2 6 q n2 ( + 3q + q n0 2 ( + 3q n + q 2n + 2 ψ( q (q4 ; q 4 3 (q 6 ; q 6 +. (3.3 3q n + q2n Entry 3.4. [9, p. 7, no. 6] With φ(q defined by (.2, 2 ( + eπi/4 φ(iq + 2 ( + e πi/4 φ( iq q n2 ( + 2q + q 2 ( + 2q n + q 2n n0 + 2 ψ( q( q 2 ; q 4 n n +. (3.4 2q n + q2n Note that the series on the right side above is f 2 (q in the notation of (2.22.

7 Mock Theta Function Identities 7 4. Proof of Entry 3. Let a 2 cos(θ, b 2 sin(θ, and t e iθ. Then, it is easy to verify that f a (q G( t, q, f a (q G(t, q, f b (q G(it, q, (4. and b a + 2 i 4 4t ( it( t, b + a i ( it( + t, 4 4t b 2 i 2t ( t2, a 2 b cos(4θ (t 4 + t 4. (4.2 Using (4. and (4.2, we can rewrite (3. as (i ( it( tg( t, q + 4t ( + i ( + t( itg(t, q 4t i 2t ( t2 G(it, q (q 4 ; q 4 ( itq (it q. (4.3 ( q; q 2 (t 4 q 4 ; q 4 (t 4 q 4 ; q 4 Multiplying both sides of (4.3 by + it, we obtain (i ( t 4 4t ( i i G( t, q G(t, q + + t t it G(it, q ( + it(q4 ; q 4 ( itq (it q ( q; q 2 (t 4 q 4 ; q 4 (t 4 q 4 ; q 4. (4.4 Using the definition (2.33 and dividing both sides of (4.4 by (i ( t 4 /(4t, we see that (3. is equivalent to the identity V ( t, q iv (t, q + (i V (it, q (q 4 ; q 4 ( + it( itq (it q 2( + it ( q; q 2 ( t 4 (t 4 q 4 ; q 4 (t 4 q 4 ; q 4 (q 4 ; q 4 ( it (it q 2( + it ( q; q 2 (t 4 ; q 4 (t 4 q 4 ; q 4 (q 4 ; q 4 2 2( + it f(it, it q ( q; q 2 (q; q f( t 4, t 4 q 4, (4.5 where in the last step we used the Jacobi triple product identity (2.9. We will verify that (4.5 is valid for q < t < q for any fixed q <. Let L(t, q :V ( t, q iv (t, q + (i V (it, q, (q 4 ; q 4 2 R(t, q : 2( + it f(it, it q ( q; q 2 (q; q f( t 4, t 4 q 4. The proof of Entry 3. will be complete once we show that R(t, q L(t, q 0. This will be achieved by showing that R(t, q L(t, q satisfies a q-difference equation of the sort stated in Lemma 2.3 and has no poles, thereby, forcing it to vanish identically.

8 8 Mock Theta Function Identities Note that if we define k(z : f(cz, c z q, then by (?? we have k(zq k(z f(czq, c z f(cz, c z q c z f(cz, c z q f(cz, c z q (cz. (4.6 Following the same reasoning of (4.6, we obtain R(tq, q R(t, q tq t f(itq, it f(it, it q f( t 4 q 4, t 4 f( t 4, t 4 q 4 q (it ( t 4 iqt3. Let us verify now that L(t, q also satisfies the same q difference equation. To that end, L(tq, q iqt 3 L(t, q V ( tq, q iv (tq, q + (i V (itq, q iqt V 3 ( t, q iv (t, q + (i V (it, q V ( tq, q qt 3 V (t, q i V (tq, q + qt 3 V ( t, q + (i V (itq, q iqt 3 V (it, q ( qt 2 i( ( q( t 2 + (i ( q(it 2 + qt 2 i( + qt 2 + (i ( + qt 2 0, where we employed (2.34. Now Lemma 2.3 implies that R(t, q V (t, q either has at least 3 poles in the region q < z, or vanishes identically. But R(t, q V (t, q has at most 3 poles, namely at t,, and i in that region, and they are all removable as we shall demonstrate. It suffices to show that t is a removable singularity. Thus, lim( tl(t lim( t V ( t, q iv (t, q + (i V (it, q t t i i lim( t G( t, q G(t, q + t + t t it G(it, q i lim G(t, q i( q; q, (4.7 t by (2.24.

9 Next, by two applications of (2.9 and (2.2, Mock Theta Function Identities 9 lim( tr(t t (q 4 ; q 4 2 lim( t 2( + it f(it, it q t ( q; q 2 (q; q f( t 4, t 4 q 4 (q 4 ; q 4 2 2( + i lim( tt f(it, it q t ( q; q 2 (q; q (t 4 ; q 4 (t 4 q 4 ; q 4 (q 4 ; q 4 (q 4 ; q 4 f(it, it q 2( + i lim( tt t ( q; q 2 (q; q ( t 4 (t 4 q 4 ; q 4 (t 4 q 4 ; q 4 ( + i(q 4 ; q 4 f(i, iq ( + if(i, iq 2( q; q 2 (q; q (q 4 ; q 4 (q 4 ; q 4 2( q; q 2 (q; q (q 4 ; q 4 ( + i( i; q (iq; q (q; q 2( q; q 2 (q; q (q 4 ; q 4 ( q 2 ; q 2 i ( q; q 2 (q 4 ; q 4 ( + i( + i( iq; q (iq; q 2( q; q 2 (q 4 ; q 4 i ( q; q 2 (q 4 ; q 4 (q 2 ; q 4 i i. (4.8 ( q; q 4 ( q 3 ; q 4 (q 4 ; q 4 (q 2 ; q 4 ( q; q Hence, by (4.7 and (4.8, L(t, q R(t, q has a removable singularity at t. By our earlier remarks, this completes the proof of Entry Proof of Entry 3.2 Our proof of Entry 3.2 is similar to our proof of Entry 3.. Since 3 a 2 +ab+b 2 (a b 2 +3ab (a + b 2 ab, we must have ab < 4. Assume without lost of generality that a < b, and let a 2 cos(θ. Solving a 2 + ab + b 2 3 for b gives b cos(θ 3 sin(θ. We will take b cos(θ + 3 sin(θ 2 sin(θ π/6, since replacing θ by θ gives the other value for b. Let t e iθ and ρ e 2πi/3. Using this parametrization we obtain which, in turn, implies that One can easily verify that a t + t, b ρ t + ρt, and a + b ρt ρ t, f a (q G(t, f b (q G(ρ t, and f a+b (q G(ρt. a + t3 t( t, b + ρ( t3 t( ρ t, and a + b ρ ( t 3 t( ρt. Now, the left side of (3.2 which we recall below, becomes (a + f a (q + (b + f b (q (a + b f a+b (q t3 t( t G(t + ρ( t3 t( ρ t G(ρ t + ρ ( t 3 t( ρt G(ρt ( t3 V (t + ρv (ρ t + ρ V (ρt. t

10 0 Mock Theta Function Identities While the right hand side of (3.2, after observing that ab(a + b 2 cos(3θ (t 3 + t 3, reduces to 3(q 3 ; q 3 2. (q; q (t 3 q 3 ; q 3 (t 3 q 3 ; q 3 Thus, Entry 3.2 is equivalent, by (2.9, to the identity V (t + ρv (ρ t + ρ V (ρt 3t(q 3 ; q 3 3 f( qf( t 3, t 3 q 3. (5. Let N(t and D(t denote the right and left sides of (5., respectively. We will verify that N(t D(t satisfies the q-difference equation N(tq D(tq qt 3 (N(t D(t without any poles in q < t. Then using Lemma 2.3, we conclude that N(t D(t 0. We employ (4.6 with c, and t and q replaced by t 3 and q 3, respectively, to deduce that Next, N(tq N(t tq t ( t 3 qt3. D(tq + qt 3 D(t V (tq + ρv (ρ tq + ρ V (ρtq + qt V 3 (t + ρv (ρ t + ρ V (ρt V (tq + qt 3 V (t + ρ V (ρ tq + qt 3 V (ρ t + ρ V (ρtq + qt 3 V (ρt qt 2 + ρ ( q(ρ t 2 + ρ ( q(ρt 2 + ρ + ρ qt 2( + ρ + ρ 0, where we used (2.34. Lemma 2.3 now implies that either N(t D(t vanishes or has 3 more poles than zeros in q < t. But N(t D(t has at most three poles, namely at t, ρ, ρ, and they are all removable as we demonstrate. It suffices to show that t is removable. By (2.24, lim( td(t lim t lim( t t lim t G(t f( q. ( t t By the Jacobi triple product identity (2.9, V (t + ρv (ρ t + ρ V (ρt t G(t + ρ ρ t G(ρ t + ρ ρt G(ρt 3t(q 3 ; q 3 3 lim( tn(t lim( t t t f( qf( t 3, t 3 q 3 3t(q 3 ; q 3 2 lim( t t f( q( t 3 (t 3 q 3 ; q 3 (t 3 q 3 ; q 3 f( q.

11 Mock Theta Function Identities We have shown that N(t D(t has a removable singularity at t. By our earlier remarks this completes the proof of Entry Proof of Entry 3.3 If a, b 3 in Entry 3., then f ( q + f ( q f 3 (q (q4 ; q 4 3q n + q 2n ( q; q 2 + q 4n + q. 8n Multiplying both sides by 2/ 3, we find that We need to show then that ψ( q (q4 ; q 4 (q 6 ; q 6 n f ( q f ( q 2 f 3 (q (q 4 ; q 4 ( q; q 2 n n 3q n + q 2n + q 4n + q 8n. + 3q n + q 2 (q 4 ; q 4 2n 3 ( q; q 2 Recall that ψ is defined by (2.. Now, (q 6 ; q 6 ( 3q n + q 2n ( + 3q n + q 2n ( q; q 2 + q 4n + q 8n n (q6 ; q 6 ( q; q 2 (q6 ; q 6 ( q; q 2 n n q 2n + q 4n + q 4n + q 8n + q 2n + q 4n (q6 ; q 6 (q 2 ; q 2 ( q; q 2 (q 6 ; q 6 (q2 ; q 2 ( q; q 2 ψ( q, n 3q n + q 2n. (6. + q 4n + q8n where in the last step (2.4 is used. The equality (6. now follows, and so the proof of Entry 3.3 is complete. 7. Proof of Entry 3.4 Let α e iπ/4. Clearly, using the notation of (2.23, we have φ(q G(i, q, and f 2 (q G( α, q. We can then restate Entry 3.4 as + α 2 G(i, iq + + α G(i, iq G( α, q ψ( q( q 2 ; q ( αq ( α q

12 2 Mock Theta Function Identities Dividing both sides by ( + α/2 and employing (2.9, we arrive at G(i, iq + α G(i, iq 2 + α G( α, q 2 ψ( qf( q( q2 ; q 4. (7. f(α, α q If we replace q by iq, (7. becomes G(i, q + α G(i, q 2V ( α, iq 2 ψ( iqf( iq(q2 ; q 4. (7.2 f(α, αq The following identities will be needed for the remainder of the proof: f(α, α qf( α, α q ( i( q 4 ; q 4 f 2 ( q, (7.3 f(α, αqf( α, αq ( i( q 4 ; q 4 f 2 ( iq, (7.4 f(α, α q ψ(iq + αψ( iq, (7.5 f( α, α q ψ(iq αψ( iq, (7.6 f(α, αq ψ( q + αψ(q, (7.7 f( α, αq ψ( q αψ(q. (7.8 We now offer proofs for all six identities. To prove (7.3 we employ (2.9 to find that f(α, α qf( α, α q ( α ( α q f( q(α (α q f( q (i; q 2 ( iq 2 ; q 2 f 2 ( q ( i( q 4 ; q 4 f 2 ( q. Clearly, (7.4 is obtained by replacing q by iq in (7.3. Recall that ψ(q f(q, q 3. From (2.7 and (2.8, f(α, α q + f( α, α q 2f(α 2 q, α 2 q 3 2f(iq, iq 3 2ψ(iq, (7.9 f(α, α q f( α, α q 2αf(α 2 q, α 2 q 3 2αf( iq, iq 3 2αψ( iq. (7.0 Equalities (7.9 and (7.0 readily imply (7.5 and (7.6. And finally we obtain (7.7 and (7.8 by replacing q by iq in (7.5 and (7.6, respectively.

13 Mock Theta Function Identities 3 We now return to (7.2 and use (7.4, (7.8, and (2.4 with q replaced by iq to deduce that It suffices now to prove (7.. Let The identity, G(i, q + α G(i, q 2V ( α, iq 2 ψ( iqf( iq(q2 ; q 4 f(α, αq 2 ψ( iqf( iq(q2 ; q 4 f( α, αq f(α, αqf( α, αq 2 ψ( iqf( iq(q2 ; q 4 (ψ( q αψ(q ( i( q 4 ; q 4 f 2 ( iq α ψ( iq(q2 ; q 4 (ψ( q αψ(q ( q 4 ; q 4 f( iq α (q2 ; q 4 (ψ( q αψ(q ( q 4 ; q 4 ( q 2 ; q 4 α (q2 ; q 4 (ψ( q αψ(q ( q 2 ; q 2 α(q 2 ; q 4 2 (ψ( q αψ(q. (7. K(t, q :αv (it, iq αv ( it, iq + iv (t, iq + iv ( t, iq + ( iv ( αt, q ( + iv (αt, q. (7.2 K(t, q 4α f 3 ( q 4 f(α t, αt q t f( iqf(α, α qf( t 4, t 4 q 4 ψ( q 2 f 2 ( qf(α t, αt q 2( + it f(t, t qf( it, it qf( it 2, it 2 q 2, (7.3 together with Entry 3. will be used to verify (7.. We will not prove (7.3, because its proof is very similar to that of (5.. The q-difference equation satisfied by K(t, q is K(tq, q αqt 3 K(t, q. It then suffices, by Lemma 2.3, to verify that the residues at four of the six singularities match those of the two representations (7.2 and (7.3 of K(t, q. It is easily verified that t α is a zero for the two representations (7.2 and (7.3 of K(t, q. Therefore, one only needs to check the residues at any three of the six singularities. If we knew the two other zeros whose existence is guaranteed by Lemma 2.3, we then would be able to reduce the right hand side of (7.3 to a single product, but we are unable to determine these two zeros. Let us define, by using (4.5, E(t, q :V ( t, q iv (t, q + (i V (it, q (7.4 (q 4 ; q 4 2 2( + it f(it, it q ( q; q 2 (q; q f( t 4, t 4 q 4. (7.5

14 4 Mock Theta Function Identities We will verify by using (7.4 and (7.2 that G(i, q + α G(i, q 2V ( α, iq ( E(α, iq + E( α, iq + α( i 2 ik(α, q + αk(α, q. (7.6 2 Equalities (7.5 and (7.3 will then be used to verify that (7.6 reduces to the right hand side of (7., which will complete the proof of Entry 3.4. Using (7.4, we have E(t, q + E( t, q ( i V (t, q + V ( t, q V (it, q V ( it, q. Setting t α, we find that E(α, q + E( α, q ( i V (α, q + V ( α, q V (iα, q V ( iα, q ( i V (α, q + V ( α, q V ( α, q V (α, q ( i V (α, q + V ( α, q αv ( α, q + αv (α, q. Replacing q by iq and dividing by α( i, we obtain By (7.2, ( E(α, iq + E( α, iq α( i α V (α, iq + α V ( α, iq V ( α, iq + V (α, iq. (7.7 K(α, q αv (iα, iq αv ( iα, iq + iv (α, iq + iv ( α, iq + ( iv ( i, q ( + iv (i, q αv ( α, iq αv (α, iq + iv (α, iq + iv ( α, iq + ( iv ( i, q ( + iv (i, q iv ( α, iq + iv (α, iq + iv (α, iq + iv ( α, iq i( iv (i, q ( + iv (i, q 2iV (α, iq + 2iV ( α, iq 2iG(i, q. (7.8

15 Combining (7.7 and (7.8, we find that Mock Theta Function Identities 5 ( E(α, iq + E( α, iq + α( i 2 ik(α, q + αk(α, q 2 α V (α, iq + α V ( α, iq V ( α, iq + V (α, iq V (α, iq V ( α, iq + G(i, q α V (α, iq α V ( α, iq + α G(i, q G(i, q + α G(i, q 2V ( α, iq. This proves our first claim that (7.6 holds. Using (7.3, (2.3, (??, and (2.20 with q replaced by q 4, we find that f 3 ( q 4 f(, q K(α, q 4 f( iqf(α, α qf(, q 4 ψ( q 2 f 2 ( qf(, q 2α( + i f(α, α qf( iα, iα qf(, q 2 f 3 ( q 4 ψ(q ψ( q 2 f 2 ( qψ(q 4 2α( + i f( iqf(α, α qψ(q 4 f(α, α qf(α, αqψ(q 2 f 3 ( q 4 ψ(q 4 f( iqf(α, α qψ(q 4 + 2( f 2 ( qψ(q iψ( q2 f 2 (α, α qψ(q 2 ϕ 2 ( q 4 ψ(q 4 f( iqf(α, α q + 2( f 2 ( qψ(qf( α, α q iψ( q2 f 2 (α, α qψ(q 2 f( α, α q. Using (7.3 and (7.6 above, we deduce that ϕ 2 ( q 4 ψ(q K(α, q 4 f( iqf(α, α q ( ψ( q 2 f 2 ( qψ(q ψ(iq αψ( iq + 2( i f(α, α qψ(q 2 ( i( q 4 ; q 4 f 2 ( q ϕ 2 ( q 4 ψ(q 4 f( iqf(α, α q + 2 ψ( q 2 ψ(qψ(iq f(α, α qψ(q 2 ( q 4 ; q 4 ψ( q 2 ψ(qψ( iq 2α f(α, α qψ(q 2 ( q 4 ; q 4 ϕ 2 ( q 4 ψ(q 4 f( iqf(α, α q + 2 ψ( q 2 ψ(qf 2 (q 2 f( iqf(α, α qψ(q 2 ( q 4 ; q 4 ψ( q 2 ψ(qf 2 (q 2 2α, (7.9 f(iqf(α, α qψ(q 2 ( q 4 ; q 4

16 6 Mock Theta Function Identities where we used (2.8 in the form f(qψ( q f 2 ( q 2 with q replaced by iq and iq, respectively. But by (2.7, ψ( q 2 f 2 (q 2 ψ(q 2 ( q 4 ; q 4 f(q2 ψ(q 2 f( q 2 ψ(q 2 ( q 4 ; q 4 ( q2 ; q 2 (q 2 ; q 2 ( q 4 ; q 4 ( q2 ; q 4 (q 4 ; q 4 (q 2 ; q 4 (q 4 ; q 4 ( q 4 ; q 4 (q4 ; q 8 (q 4 ; q 4 2 ( q 4 ; q 4 (q4 ; q 4 2 ( q 4 ; q 4 2 ϕ 2 ( q 4, (7.20 where we used Euler s identity (2.2, and (2.5. Using (7.20 in (7.9, (2.8 in the form f(qψ( q f 2 ( q 2 with q replaced by iq and iq, respectively, (7.5, and (2.5, we deduce that ϕ 2 ( q 4 ψ(q K(α, q 4 f( iqf(α, α q + 2 ϕ 2 ( q 4 ψ(q f( iqf(α, α q 2α ϕ2 ( q 4 ψ(q f(iqf(α, α q 2 ϕ2 ( q 4 ψ(q ( f(α, α q f( iq + α f(iq 2 ϕ2 ( q 4 ψ(q ( ψ(iq + αψ( iq f(α, α qf 2 (q 2 2 ϕ2 ( q 4 ψ(q (q 4 ; q ψ(q f 2 (q 2 ( q 4 ; q 4 2 ( q 2 ; q 2 2 (q 4 ; q ψ(q ψ(q 2 ( q 4 ; q 4 2 ( q 2 ; q 4 2 (q 4 ; q 4 2 ( q 2 ; q 2 2 2(q 2 ; q 4 2 ψ(q, (7.2 where in the last step we used (2.2. Thus, by (7.2, 2 ik(α, q + 2 αk(α, q i(q2 ; q 4 2 ψ(q α(q 2 ; q 4 2 ψ( q. (7.22 Let us evaluate now E(α, q. By (7.5, (??, and (2.3, E(α, q 2( + iα (q4 ; q 4 2 f(iα, iα q (q; q ( q; q 2 f(, q 4 2( + iα (q4 ; q 4 2 f( α, αq (q; q ( q; q 2 f(, q 4 ( + i (q4 ; q 4 2 f( α, α q (q; q ( q; q 2 ψ(q 4. (7.23

17 Employ (2. and (2.2 to deduce that (q 4 ; q 4 2 (q; q ( q; q 2 ψ(q 4 Mock Theta Function Identities 7 (q 4 ; q 4 2 (q 2 ; q 2 (q; q 2 ( q; q 2 ( q 4 ; q 4 2 (q 4 ; q 4 (q 4 ; q 4 (q 2 ; q 2 (q 2 ; q 4 ( q 4 ; q 4 2 (q2 ; q 2 ( q 2 ; q 2 2 (q 2 ; q 2 ( q 4 ; q 4 2 Using (7.24 in (7.23, we obtain Similarly, we obtain ( q2 ; q 2 2 ( q 4 ; q 4 2 Combining (7.25 and (7.26, we arrive at ( q2 ; q 4 2 ( q 4 ; q 4 2 ( q 4 ; q 4 2 ( q 2 ; q 4 2. (7.24 E(α, q ( + i( q 2 ; q 4 2 f( α, α q. (7.25 E( α, q ( + i( q 2 ; q 4 2 f(α, α q. (7.26 E(α, q + E( α, q α( i + i ( α( i ( q2 ; q 4 2 f(α, α q + f( α, α q by (7.9. Finally, replacing q by iq in (7.27, we deduce that 2α( q 2 ; q 4 2 ψ(iq, (7.27 α( i E(α, iq + E( α, iq 2α(q2 ; q 4 2 ψ( q. (7.28 Adding (7.22 and (7.28 together, we find that (7.6 reduces to the right hand side of (7., i.e., 2 ik(α, q + 2 αk(α, q + E(α, iq + E( α, iq α( i i(q 2 ; q 4 2 ψ(q α(q 2 ; q 4 2 ψ( q + 2α(q 2 ; q 4 2 ψ( q α(q 2 ; q 4 2 (ψ( q αψ(q. This completes the verification of (7., since we have already verified (7.6. Hence, the proof of Entry 3.4 is complete. 8. Proof of.8 Let us recall equation (5. and (2.36, which is the equivalent form of Entry 3.2. Thus, V (t + ρv (ρ t + ρ 3t(q 3 ; q 3 3 V (ρt (q f( t 3, t 3 q 3, (8. V (t + t ( n q3n(n+/2 (q tq, (8.2 n n

18 8 Mock Theta Function Identities where ρ e 2πi/3. Using (8.2 in (8., we obtain Then, we have 3t(q 3 ; q 3 3 (q f( t 3, t 3 q 3 + t ( n q3n(n+/2 (q tq n n + ρ + ρt ( n q3n(n+/2 (q ρ tq n n + ρ + ρ t ( n q3n(n+/2 (q ρtq n n t ( n q 3n(n+/2 (q tq + ρ n ρ tq + ρ n ρtq n n 3t ( n q3n(n+/2 (q t 3 q. 3n n (q 3 ; q 3 3 f( t 3, t 3 q 3 n ( n q3n(n+/2. (8.3 t 3 q3n Now, (.8 follows if one replaces q 3 by q and t 3 by t, respectively, and employs (2.9 in (8.3. Acknowledgment. I would like to thank my adviser Professor Bruce C. Berndt for his guidance and assistance at all stages of this work. References [] G. E. Andrews, On basic hypergeometric series, mock theta functions, and partitions. I, Quart. J. Math. Oxford Ser. (2 7 (966, [2] G. E. Andrews, Mock theta functions. Theta functions Bowdoin 987, Part 2 (Brunswick, ME, 987, Proc. Sympos. Pure Math. 49, pp [3] A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London. Math. Soc. (3 4 (954, [4] B. C. Berndt, Ramanujan s Notebooks, Part III, Springer Verlag, New York, 99. [5] F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge 8 (944, 0 5. [6] R. J. Evans, Generalized Lambert series, in Analytic Number Theory (Allerton Park, IL, 995, Vol., B. C. Berndt, H. G. Diamond, and A. J. Hildebrand, eds., Birkhäuser, Boston, 996, pp [7] N. J. Fine, Basic Hypergeometric Series and Applications, Mathematical Surveys and Monographs, 27, American Mathematical Society, Providence, RI, 988. [8] F. G. Garvan, New combinatorial interpretations of Ramanujan s partition congruences mod 5, 7 and, Trans. Amer. Math. Soc. 305 (988, [9] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 988. [0] G. N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc. (936,

19 Mock Theta Function Identities 9 Department of Mathematics, University of Illinois, 409 West Green Street, Urbana, IL 680, USA address: yesilyur@math.uiuc.edu