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Partitions and powers of 13
Accepted Manuscript Partitions and powers of 13
Michael D. Hirschhorn
PII: DOI: Reference:
S0022-314X(17)30129-4 http://dx.doi.org/10.1016/j.jnt.2017.02.016 YJNTH 5720
To appear in:
Journal of Number Theory
Received date: Accepted date:
7 October 2016 6 February 2017
Please cite this article in press as: M.D. Hirschhorn, Partitions and powers of 13, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2017.02.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal of Number Theory
Ramanujan partition congruence modulo 11 Journal of Number Theory 00 (2017) 1–9
Partitions and powers of 13 Michael D. Hirschhorn School of Mathematics and Statistics, UNSW, Sydney 2052, Australia
Abstract In 1919, Ramanujan gave the identities
p(5n + 4)qn = 5
n≥0
and
(1 − q5n )5 (1 − qn )6 n≥1
(1 − q7n )3 (1 − q7n )7 + 49q n 4 (1 − q ) (1 − qn )8 n≥0 n≥1 n≥1 and in 1939, H.S. Zuckerman gave similar identities for p(25n + 24)qn , p(49n + 47)qn and p(13n + 6)qn .
p(7n + 5)qn = 7
n≥0
n≥0
n≥0
From Zuckerman’s paper, it would seem that this last identity is an isolated curiosity, but that is not the case. Just as the first four mentioned identities are well known to be the earliest instances of infinite families of such identities for powers of 5 and 7, the fifth identity is likewise the first of an infinite family of such identities for powers of 13. We will establish this fact and give the second identity in the infinite family. c 2011 Published by Elsevier Ltd.
1. Introduction Throughout the paper, it should be understood that E(q) is Euler’s product, (1 − qn ). E(q) = n≥1
In 1919, Ramanujan [4] essentially proved that
E(q5 )5 E(q)6
(1)
E(q7 )3 E(q7 )7 + 49q . 4 E(q) E(q)8
(2)
p(5n + 4)qn = 5
n≥0
and stated without proof that n≥0
p(7n + 5)qn = 7
1
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
2
In 1939, H.S. Zuckerman [5] went one further, and proved that
p(25n + 24)qn = 63 × 52
n≥0
5 12 5 18 E(q5 )6 5 E(q ) 7 2 E(q ) + 52 × 5 q + 63 × 5 q E(q)7 E(q)13 E(q)19
E(q5 )24 E(q5 )30 + 512 q4 , 25 E(q)31 E(q) 7 8 E(q7 )4 4 E(q ) p(49n + 47)qn = 2546 × 72 + 48934 × 7 q E(q)9 E(q)5 n≥0 +6 × 510 q3
E(q7 )12 E(q7 )16 + 24888000 × 77 q3 13 E(q) E(q)17 7 20 7 24 E(q ) 11 5 E(q ) +2394438 × 79 q4 + 1437047 × 7 q E(q)21 E(q)25 7 28 E(q ) E(q7 )32 +4043313 × 712 q6 + 161744 × 715 q7 29 E(q) E(q)33 E(q7 )36 E(q7 )40 +32136 × 717 q8 + 31734 × 718 q9 37 E(q) E(q)41 7 44 7 48 E(q ) 22 11 E(q ) +3120 × 720 q10 + 204 × 7 q E(q)49 E(q)45 7 52 7 56 E(q ) E(q ) +8 × 724 q12 + 725 q13 E(q)53 E(q)57
(3)
+1418989 × 75 q2
(4)
as well as
p(13n + 6)qn = 11
n≥0
13 5 E(q13 ) E(q13 )3 2 2 E(q ) + 36 × 13q + 38 × 13 q E(q)2 E(q)4 E(q)6
E(q13 )7 E(q13 )9 E(q13 )11 + 6 × 134 q4 + 135 q5 8 10 E(q) E(q) E(q)12 13 13 E(q ) +135 q6 . E(q)14 +20 × 133 q3
(5)
It is well-known that the identities (1)–(4) are the earliest cases of infinite families of identities for powers of 5 and 7. What is not well-known is that Zuckerman’s identity (5) is likewise the first of an infinite family of similar identities for powers of 13. We demonstrate this fact and give the second identity in the infinite family. Related matters are discussed in [1]. In particular, the modular equation D = 0 (D is defined in equation (20)), is given in [1], Appendix C. 2. The infinite family of identities for powers of 13 We will show that the following holds. For α ≥ 0, (132α+2 −1)/24 11 × 132α+1 + 1 n E(q13 )2i−1 q = p 132α+1 n + x2α+1,i qi−1 24 E(q)2i n≥0 i=1 2
(6)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
3
and (132α+3 −13)/24 E(q13 )2i 23 × 132α+2 + 1 n p 132α+2 n + x2α+2,i qi−1 . q = 24 E(q)2i+1 n≥0 i=1
(7)
where the coefficient vectors xα = (xα,i )i≥1 are given recursively by x1 = (11, 36 × 13, 38 × 132 , 20 × 133 , 6 × 134 , 135 , 135 , 0, · · · )
(8)
and for α ≥ 0, x2α+2 = x2α+1 A, x2α+3 = x2α+2 B,
(9) (10)
where A = (ai, j )i, j≥1 and B = (bi, j )i, j≥1 are defined by ai, j = m2i,i+ j and bi, j = m2i+1,i+ j
(11)
where the mi, j are given by mi, j xi y j i, j≥1
= (11y + 36 × 13y2 + 38 × 132 y3 + 20 × 135 y4 + 6 × 134 y5 + 135 y6 + 135 y7 )x −(408y2 + 692 × 13y3 + 444 × 132 y4 + 148 × 133 y5 + 2 × 135 y6 + 2 × 135 y7 )x2 +(108y2 + 378 × 13y3 + 306 × 132 y4 + 114 × 133 y5 + 21 × 134 y6 + 21 × 134 y7 )x3 −(1384y3 + 1688 × 13y4 + 737 × 132 y5 + 148 × 133 y6 + 12 × 134 y7 )x4 +(190y3 + 510 × 13y4 + 280 × 132 y5 + 5 × 134 y6 + 75 × 133 y7 )x5 −(1332y4 + 1104 × 13y5 + 306 × 132 y6 + 30 × 133 y7 )x6 +(140y4 + 266 × 13y5 + 7 × 133 y6 + 133 × 132 y7 )x7 −(592y5 + 296 × 13y6 + 40 × 132 y7 )x8 + (54y5 + 63 × 13y6 + 135 × 13y7 )x9 −(10 × 13y6 + 30 × 13y7 )x10 + (11y6 + 77y7 )x11 − 12y7 x12 + y7 x13
1 − (11 × 13y + 36 × 132 y2 + 38 × 133 y3 + 20 × 134 y4 + 6 × 135 y5 + 136 y6 + 136 y7 )x +(204 × 13y2 + 346 × 132 y3 + 222 × 133 y4 + 74 × 134 y5 + 136 y6 + 136 y7 )x2 −(36 × 13y2 + 126 × 132 y3 + 102 × 133 y4 + 38 × 134 y5 + 7 × 135 y6 + 7 × 135 y7 )x3 +(346 × 13y3 + 422 × 132 y4 + 184 × 133 y5 + 37 × 134 y6 + 3 × 135 y7 )x4 −(38 × 13y3 + 102 × 132 y4 + 56 × 133 y5 + 135 y6 + 15 × 134 y7 )x5 +(222 × 13y4 + 184 × 132 y5 + 51 × 133 y6 + 5 × 134 y7 )x6 −(20 × 13y4 + 38 × 132 y5 + 134 y6 + 19 × 133 y7 )x7 +(74 × 13y5 + 37 × 132 y6 + 5 × 133 y7 )x8 − (6 × 13y5 + 7 × 132 + 15 × 132 y7 )x9 +(132 y6 + 3 × 132 y7 )x10 − (13y6 + 7 × 13y7 )x11 + 13y7 x12 − y7 x13 .
3. Proof The following results were stated by Hirschhorn [3]. 3
(12)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
4
If ζ and T are defined by ζ=
E(q) E(q13 )2 and T = q7 E(q169 ) q13 E(q169 )2
(13)
and if H is the huffing operator, given by ⎛ ⎞ ⎜⎜⎜ ⎟ n⎟ ⎜ H ⎜⎝ c(n)q ⎟⎟⎟⎠ = c(13n)q13n , n
(14)
n
then H(ζ) = 1, H(ζ 2 ) = −2T − 1, H(ζ 3 ) = 13, H(ζ 4 ) = 2T 2 − 13, H(ζ 5 ) = −20T 2 − 10 × 13T − 132 , H(ζ 6 ) = 10T 3 − 132 , H(ζ 7 ) = 98T 3 + 28 × 13T 2 − 133 , H(ζ 8 ) = −70T 4 − 133 , H(ζ 9 ) = −162T 4 + 108 × 13T 3 + 72 × 132 T 2 + 18 × 133 T + 134 , H(ζ 10 ) = 288T 5 − 134 , H(ζ 11 ) = −902 × T 5 − 1672 × 13T 4 − 792 × 132 T 3 −198 × 133 T 2 − 22 × 134 T − 135 , H(ζ 12 ) = −418 × T 6 − 135 .
(15)
Let η be a 13-th root of unity other than 1 and for k = 0, · · · , 12, let ζk = ζ(ηk q).
(16)
Then [3] the symmetric functions of the ζk are σ1 = ζk = 13, k
σ2 =
ζk ζl = 13T + 7 × 13,
k
σ3 =
ζk ζl ζm = 132 T + 3 × 132 ,
k
σ4 = 6 × 13T 2 + 7 × 132 T + 15 × 132 , σ5 = 74 × 13T 2 + 37 × 132 T + 5 × 133 , σ6 = 20 × 13T 3 + 38 × 132 T 2 + 134 T + 19 × 133 , σ7 = 222 × 13T 3 + 184 × 132 T 2 + 51 × 133 T + 5 × 134 , σ8 = 38 × 13T 4 + 102 × 132 T 3 + 56 × 133 T 2 + 135 T + 15 × 134 , σ9 = 346 × 13T 4 + 422 × 132 T 3 + 184 × 133 T 2 + 37 × 134 T + 3 × 135 , σ10 = 36 × 13T 5 + 126 × 132 T 4 + 102 × 133 T 3 + 38 × 134 T 2 + 7 × 135 T + 7 × 135 , σ11 = 204 × 13T 5 + 346 × 132 T 4 + 222 × 133 T 3 + 74 × 134 T 2 + 136 T + 136 σ12 = 11 × 13T 6 + 36 × 132 T 5 + 38 × 133 T 4 + 20 × 134 T 3 + 6 × 135 T 2 + 136 T + 136 and σ13 =
k
ζk =
E(q13 )14 q91 E(q169 )14 4
= T 7.
(17)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
We now determine the generating function i≥1
We have
H
i≥1
H
1 i x. ζi
⎛ ⎞ 1 i 1 ⎜⎜⎜ 1 1 1 ⎟⎟⎟ i ⎜ ⎟x x = + + · · · + ⎝ i ⎠ ζi 13 ζ0i ζ1i ζ12 i≥1 x x 1 x + + ··· + = 13 ζ0 − x ζ1 − x ζ12 − x N = , D
5
(18)
(19)
where D = (ζ0 − x)(ζ1 − x) · · · (ζ12 − x) = σ13 − σ12 x + σ11 x2 − + · · · + σ1 x12 − x13 = T 7 − (11 × 13T 6 + 36 × 132 T 5 + · · · + 136 )x + − · · · + 13x12 − x13
(20)
and x (ζl − x) 13 k=0 lk 12
N=
x ∂D 13 ∂x = (11T 6 + 36 × 13T 5 + · · · + 135 )x − + · · · − 12x12 + x13 .
=−
If we divide numerator and denominator by T 7 , we find 1 11 135 x − + · · · + 7 x13 + · · · + T T7 T 1 i H i x = . ζ 136 11 × 13 1 i≥1 + · · · + 7 x + − · · · − 7 x13 1− T T T It follows from this that H
mi, j 1 , = ζi Tj j≥1
(21)
(22)
(23)
where mi, j = 0 f or j > 7i
(24)
m2i, j = 0 f or j < i and m2i+1, j = 0 f or j < i.
(25)
and
Indeed, we have
135 1 11 + · · · + 7 x − + · · · + 7 x13 mi, j T T T xi = . j T 136 11 × 13 1 i≥1 j≥1 + · · · + 7 x + − · · · − 7 x13 1− T T T 5
(26)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
By the simple expedient of replacing
6
1 by y in (26), we obtain (12). T
From (23)–(25), we obtain 1 11 36 × 13 38 × 132 20 × 133 6 × 134 135 135 H = + + + + + 6 + 7, ζ T T2 T3 T4 T T5 T
(27)
and, for i ≥ 1, H
14i 13i 13i mi, j m2i,i+ j ai, j 1 = = = j i+ j 2i T T T i+ j ζ j=i+1 j=1 j=1
(28)
and H
1
=
ζ 2i+1
14i+7
m2i+1,i+ j 13i+7 bi, j m2i+1, j 13i+7 = = , j i+ j T T T i+ j j=i+1 j=1 j=1
(29)
where the ai, j and bi, j satisfy (11). Now, (27) becomes 7 39 169 6 q E(q169 ) q26 E(q169 )4 ) q13 E(q169 )2 2 q E(q H + 36 × 13 + 38 × 13 = 11 13 2 13 4 13 6 E(q) E(q ) E(q ) E(q ) +20 × 134
65 169 10 78 169 12 91 169 14 q52 E(q169 )8 ) ) ) 4 q E(q 5 q E(q 5 q E(q + 6 × 13 + 13 + 13 , E(q13 )8 E(q13 )10 E(q13 )12 E(q13 )14
(30)
and this reduces easily to (5). (28) becomes ⎛ 13 i+ j ⎞ 13i ⎜⎜⎜ q7 E(q169 ) 2i ⎟⎟⎟ q E(q169 )2 ⎟⎟⎠ = ⎜ ai, j , H ⎜⎝ E(q) E(q13 )2 j=1
(31)
or,
13 2i−1
13i
E(q169 )2 j , E(q13 )2 j+1
(32)
⎛ 13 i+ j ⎞ ⎜⎜⎜ q7 E(q169 ) 2i+1 ⎟⎟⎟ 13i+7 q E(q169 )2 ⎟⎟⎠ = ⎜ bi, j , H ⎜⎝ E(q) E(q13 )2 j=1
(33)
) E(q)2i
i−13 E(q
H q
=
j=1
ai, j q13 j−13
while (29) becomes
or, i−6
H q
13i+7 169 2 j−1 E(q13 )2i ) 13 j−13 E(q b q . = i, j 13 )2 j E(q)2i+1 E(q j=1
(34)
Now we prove (6) and (7) by induction. We have seen that (5) holds. This is the case α = 0 of (6). Suppose (6) holds for some α ≥ 0. Then (132α+2 −1)/24 E(q13 )2i−1 11 × 132α+1 + 1 n−12 2α+1 q p 13 n+ = x2α+1,i qi−13 . 24 E(q)2i n≥0 i=1 6
(35)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
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If we apply H to (35), we find
11 × 132α+1 + 1 13n p 132α+1 (13n + 12) + q 24 n≥0 (132α+2 −1)/24
13i
E(q169 )2 j E(q13 )2 j+1 i=1 j=1 ⎛ ⎞ (132α+3 −13)/24 ⎜(132α+2 −1)/24 ⎟⎟⎟ ⎜⎜⎜ E(q169 )2 j ⎜⎜⎜ = x2α+1,i ai, j ⎟⎟⎟⎟⎠ q13 j−13 ⎝ E(q13 )2 j+1 j=1 i=1 =
=
x2α+1,i
(132α+3 −13)/24
ai, j q13 j−13
x2α+2, j q13 j−13
j=1
E(q169 )2 j , E(q13 )2 j+1
(36)
or, (132α+3 −13)/24 E(q13 )2 j 23 × 132α+2 + 1 n 2α+2 q = p 13 n+ x2α+2, j q j−1 , 24 E(q)2 j+1 n≥0 j=1
(37)
which is (7). Now suppose (7) holds for some α ≥ 0. Then (132α+3 −13)/24 23 × 132α+2 + 1 n−5 E(q13 )2i 2α+2 p 13 n+ x2α+2,i qi−6 . q = 24 E(q)2i+1 n≥0 i=1
(38)
If we apply H to (38), we find
23 × 132α+2 + 1 13n q p 132α+2 (13n + 5) + 24 n≥0 (132α+3 −13)/24
13i+7
E(q169 )2 j−1 E(q13 )2 j i=1 j=1 ⎛ ⎞ (132α+4 −1)/24 ⎜(132α+3 −13)/24 ⎟⎟⎟ 169 2 j−1 ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟⎟ q13 j−13 E(q ) = x b 2α+2,i i, j ⎝ ⎠ E(q13 )2 j j=1 i=1 =
=
(132α+4 −1)/24
x2α+2,i
bi, j q13 j−13
x2α+3, j q13 j−13
j=1
E(q169 )2 j−1 , E(q13 )2 j
(39)
or, (132α+4 −1)/24 11 × 132α+3 + 1 n E(q13 )2 j−1 q = p 132α+3 n + x2α+3, j q j−1 , 24 E(q)2 j n≥0 j=1
(40)
which is (6) with α + 1 for α. This completes our proof of (6) and (7). 4. The second identity in the family
n≥0
p(169n + 162)qn =
91 i=1
7
x2,i qi−1
E(q13 )2i E(q)2i+1
(41)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
where x2 = (x2,i )i≥1 = (129913904637, 6061605043150208 × 13, 13608112586256178046 × 132 6280902061708574674384 × 133 , 80635921883086425364027 × 135 , 84554753069155774658709629 × 135 , 3916468033999952967228746399 × 136 , 116548389829851820637604548142 × 137 , 2408891668264357573695348194716 × 138 , 36596898089679538239041805817046 × 139 , 426530964811783728667060279961675 × 1310 , 3942226951585332362350072435077528 × 1311 , 385658958461803612492956251132861773 × 1311 , 185683085939890676806684250771459434 × 1313 , 983797065112634408871060753747525360 × 1314 , 4477389554472709218367190024998232130 × 1315 , 17721739170007369591672941408346874421 × 1316 , 4742355490178921142892419097596618355 × 1318 , 190225453599141520922788551641691242508 × 1318 , 524721368801814092141453950047818477326 × 1319 , 1302911978265864071115800136841270884325 × 1320 , 2929975138807498636693119222491450961566 × 1321 , 5999389868465532788358368350259321205163 × 1322 , 11238726553479300068613150748539443879990 × 1323 , 19343912101567621453437589381740398385615 × 1324 , 399199887453496794280385245259602986657786 × 1324 , 45114569049056244774820332010258592196403 × 1326 , 61531718773137637822015907451354660369102 × 1327 , 78128730639946286724152148516203955404966 × 1328 , 92588134761861713785091033497567560642309 × 1329 , 7895718682337867643451727088301900325822 × 1331 , 106674774717034569765482744541776413982076 × 1331 , 104127690533250094738334370281080119752897 × 1332 , 95632615339178385499866474240356595035734 × 1333 , 82769860945855840879428760727969455676949 × 1334 , 67607586521876860960757233146365406440864 × 1335 , 52185630770144277441328235524086183756249 × 1336 , 38112125284602081354120996414789327371490 × 1337 , 342730657709309329959063447557419449170283 × 1337 , 17291161864563075008390801545579770606804 × 1339 , 10762180018683300577592515849620716685170 × 1340 , 6361993276065687616569518872990897763233 × 1341 , 3574560685274501116380983329127804315499 × 1342 , 146936993657441362564501166152594688776 × 1344 , 8
8
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
9
971409646247113487173405170610660948070 × 1344 , 470358969516921694951165237588782471677 × 1345 , 216944721415349512912594980889327147413 × 1346 , 95351253212840170160691532583542843424 × 1347 , 39948335504280947062152709269989049847 × 1348 , 15957914357030470689632293966446907145 × 1349 , 6079141248603201845764987330492593982 × 1350 , 28714246146489509001585060775229857512 × 1350 , 765494941301775657755760752212309475 × 1352 , 253052852451392618013575739199077679 × 1353 , 79788127925215402822125633710674827 × 1354 , 23992399482226475885195013027244418 × 1355 , 529171510382781913451554617989308 × 1357 , 1880313225394592406581074643938328 × 1357 , 489788423819708495595161861618881 × 1358 , 121536611116247430071845283921984 × 1359 , 28716099109510015333499605348975 × 1360 , 6456958185273171048474896073300 × 1361 , 1380835433937604032219951482224 × 1362 , 280643142907342084675152233134 × 1363 , 704129708359731486328469591897 × 1363 , 762886312702089600644337633 × 1366 , 1721000064243705676349614170 × 1366 , 282702871665192373362871298 × 1367 , 43901108707026096873874183 × 1368 , 6435317579777653828753163 × 1369 , 888969653957555557187503 × 1370 , 115507086031646435494256 × 1371 , 14086827856186927900879 × 1372 , 1608638921698649370919 × 1373 , 171538848960796539961 × 1374 , 17028556255537068094 × 1375 , 20384714589175433972 × 1375 , 1734110498546299875 × 1376 , 135634467981532940 × 1377 , 9699744171620664 × 1378 , 630048041950131 × 1379 , 36877266904170 × 1380 , 11397690381 × 1383 , 88708802688 × 1382 , 3547018629 × 1383 , 120668870 × 1384 , 3396925 × 1385 , 75992 × 1386 , 1267 × 1387 , 14 × 1388 , 1388 , 0, · · ·). (42) [1] A.O.L. Atkin and J. N. O’Brien, Some properties of p(n) and c(n) modulo powers of 13, Trans. Amer. Math. Soc. 126 (1967), 442–459. [2] G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, eds. Ramanujan’s Collected Papers, AMS Chelsea 2000. [3] M.D. Hirschhorn, Ramanujan’s tau function, Proceedings of Alladi 60, A conference held in honor of Krishna Alladi on the occasion of his 60th Birthday, Springer, to appear. [4] S. Ramanujan, Some properties of p(n), The number of partitions of n, Proc. Camb. Philos. Soc. XIX (1919), 207–210. [5] H. S. Zuckerman, Identities analogous to Ramanujan’s identities involving the partition function, Duke Math. J. 5 (1) (1939), 88–110.
9
Michael D. Hirschhorn
PII: DOI: Reference:
S0022-314X(17)30129-4 http://dx.doi.org/10.1016/j.jnt.2017.02.016 YJNTH 5720
To appear in:
Journal of Number Theory
Received date: Accepted date:
7 October 2016 6 February 2017
Please cite this article in press as: M.D. Hirschhorn, Partitions and powers of 13, J. Number Theory (2017), http://dx.doi.org/10.1016/j.jnt.2017.02.016
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Journal of Number Theory
Ramanujan partition congruence modulo 11 Journal of Number Theory 00 (2017) 1–9
Partitions and powers of 13 Michael D. Hirschhorn School of Mathematics and Statistics, UNSW, Sydney 2052, Australia
Abstract In 1919, Ramanujan gave the identities
p(5n + 4)qn = 5
n≥0
and
(1 − q5n )5 (1 − qn )6 n≥1
(1 − q7n )3 (1 − q7n )7 + 49q n 4 (1 − q ) (1 − qn )8 n≥0 n≥1 n≥1 and in 1939, H.S. Zuckerman gave similar identities for p(25n + 24)qn , p(49n + 47)qn and p(13n + 6)qn .
p(7n + 5)qn = 7
n≥0
n≥0
n≥0
From Zuckerman’s paper, it would seem that this last identity is an isolated curiosity, but that is not the case. Just as the first four mentioned identities are well known to be the earliest instances of infinite families of such identities for powers of 5 and 7, the fifth identity is likewise the first of an infinite family of such identities for powers of 13. We will establish this fact and give the second identity in the infinite family. c 2011 Published by Elsevier Ltd.
1. Introduction Throughout the paper, it should be understood that E(q) is Euler’s product, (1 − qn ). E(q) = n≥1
In 1919, Ramanujan [4] essentially proved that
E(q5 )5 E(q)6
(1)
E(q7 )3 E(q7 )7 + 49q . 4 E(q) E(q)8
(2)
p(5n + 4)qn = 5
n≥0
and stated without proof that n≥0
p(7n + 5)qn = 7
1
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
2
In 1939, H.S. Zuckerman [5] went one further, and proved that
p(25n + 24)qn = 63 × 52
n≥0
5 12 5 18 E(q5 )6 5 E(q ) 7 2 E(q ) + 52 × 5 q + 63 × 5 q E(q)7 E(q)13 E(q)19
E(q5 )24 E(q5 )30 + 512 q4 , 25 E(q)31 E(q) 7 8 E(q7 )4 4 E(q ) p(49n + 47)qn = 2546 × 72 + 48934 × 7 q E(q)9 E(q)5 n≥0 +6 × 510 q3
E(q7 )12 E(q7 )16 + 24888000 × 77 q3 13 E(q) E(q)17 7 20 7 24 E(q ) 11 5 E(q ) +2394438 × 79 q4 + 1437047 × 7 q E(q)21 E(q)25 7 28 E(q ) E(q7 )32 +4043313 × 712 q6 + 161744 × 715 q7 29 E(q) E(q)33 E(q7 )36 E(q7 )40 +32136 × 717 q8 + 31734 × 718 q9 37 E(q) E(q)41 7 44 7 48 E(q ) 22 11 E(q ) +3120 × 720 q10 + 204 × 7 q E(q)49 E(q)45 7 52 7 56 E(q ) E(q ) +8 × 724 q12 + 725 q13 E(q)53 E(q)57
(3)
+1418989 × 75 q2
(4)
as well as
p(13n + 6)qn = 11
n≥0
13 5 E(q13 ) E(q13 )3 2 2 E(q ) + 36 × 13q + 38 × 13 q E(q)2 E(q)4 E(q)6
E(q13 )7 E(q13 )9 E(q13 )11 + 6 × 134 q4 + 135 q5 8 10 E(q) E(q) E(q)12 13 13 E(q ) +135 q6 . E(q)14 +20 × 133 q3
(5)
It is well-known that the identities (1)–(4) are the earliest cases of infinite families of identities for powers of 5 and 7. What is not well-known is that Zuckerman’s identity (5) is likewise the first of an infinite family of similar identities for powers of 13. We demonstrate this fact and give the second identity in the infinite family. Related matters are discussed in [1]. In particular, the modular equation D = 0 (D is defined in equation (20)), is given in [1], Appendix C. 2. The infinite family of identities for powers of 13 We will show that the following holds. For α ≥ 0, (132α+2 −1)/24 11 × 132α+1 + 1 n E(q13 )2i−1 q = p 132α+1 n + x2α+1,i qi−1 24 E(q)2i n≥0 i=1 2
(6)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
3
and (132α+3 −13)/24 E(q13 )2i 23 × 132α+2 + 1 n p 132α+2 n + x2α+2,i qi−1 . q = 24 E(q)2i+1 n≥0 i=1
(7)
where the coefficient vectors xα = (xα,i )i≥1 are given recursively by x1 = (11, 36 × 13, 38 × 132 , 20 × 133 , 6 × 134 , 135 , 135 , 0, · · · )
(8)
and for α ≥ 0, x2α+2 = x2α+1 A, x2α+3 = x2α+2 B,
(9) (10)
where A = (ai, j )i, j≥1 and B = (bi, j )i, j≥1 are defined by ai, j = m2i,i+ j and bi, j = m2i+1,i+ j
(11)
where the mi, j are given by mi, j xi y j i, j≥1
= (11y + 36 × 13y2 + 38 × 132 y3 + 20 × 135 y4 + 6 × 134 y5 + 135 y6 + 135 y7 )x −(408y2 + 692 × 13y3 + 444 × 132 y4 + 148 × 133 y5 + 2 × 135 y6 + 2 × 135 y7 )x2 +(108y2 + 378 × 13y3 + 306 × 132 y4 + 114 × 133 y5 + 21 × 134 y6 + 21 × 134 y7 )x3 −(1384y3 + 1688 × 13y4 + 737 × 132 y5 + 148 × 133 y6 + 12 × 134 y7 )x4 +(190y3 + 510 × 13y4 + 280 × 132 y5 + 5 × 134 y6 + 75 × 133 y7 )x5 −(1332y4 + 1104 × 13y5 + 306 × 132 y6 + 30 × 133 y7 )x6 +(140y4 + 266 × 13y5 + 7 × 133 y6 + 133 × 132 y7 )x7 −(592y5 + 296 × 13y6 + 40 × 132 y7 )x8 + (54y5 + 63 × 13y6 + 135 × 13y7 )x9 −(10 × 13y6 + 30 × 13y7 )x10 + (11y6 + 77y7 )x11 − 12y7 x12 + y7 x13
1 − (11 × 13y + 36 × 132 y2 + 38 × 133 y3 + 20 × 134 y4 + 6 × 135 y5 + 136 y6 + 136 y7 )x +(204 × 13y2 + 346 × 132 y3 + 222 × 133 y4 + 74 × 134 y5 + 136 y6 + 136 y7 )x2 −(36 × 13y2 + 126 × 132 y3 + 102 × 133 y4 + 38 × 134 y5 + 7 × 135 y6 + 7 × 135 y7 )x3 +(346 × 13y3 + 422 × 132 y4 + 184 × 133 y5 + 37 × 134 y6 + 3 × 135 y7 )x4 −(38 × 13y3 + 102 × 132 y4 + 56 × 133 y5 + 135 y6 + 15 × 134 y7 )x5 +(222 × 13y4 + 184 × 132 y5 + 51 × 133 y6 + 5 × 134 y7 )x6 −(20 × 13y4 + 38 × 132 y5 + 134 y6 + 19 × 133 y7 )x7 +(74 × 13y5 + 37 × 132 y6 + 5 × 133 y7 )x8 − (6 × 13y5 + 7 × 132 + 15 × 132 y7 )x9 +(132 y6 + 3 × 132 y7 )x10 − (13y6 + 7 × 13y7 )x11 + 13y7 x12 − y7 x13 .
3. Proof The following results were stated by Hirschhorn [3]. 3
(12)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
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If ζ and T are defined by ζ=
E(q) E(q13 )2 and T = q7 E(q169 ) q13 E(q169 )2
(13)
and if H is the huffing operator, given by ⎛ ⎞ ⎜⎜⎜ ⎟ n⎟ ⎜ H ⎜⎝ c(n)q ⎟⎟⎟⎠ = c(13n)q13n , n
(14)
n
then H(ζ) = 1, H(ζ 2 ) = −2T − 1, H(ζ 3 ) = 13, H(ζ 4 ) = 2T 2 − 13, H(ζ 5 ) = −20T 2 − 10 × 13T − 132 , H(ζ 6 ) = 10T 3 − 132 , H(ζ 7 ) = 98T 3 + 28 × 13T 2 − 133 , H(ζ 8 ) = −70T 4 − 133 , H(ζ 9 ) = −162T 4 + 108 × 13T 3 + 72 × 132 T 2 + 18 × 133 T + 134 , H(ζ 10 ) = 288T 5 − 134 , H(ζ 11 ) = −902 × T 5 − 1672 × 13T 4 − 792 × 132 T 3 −198 × 133 T 2 − 22 × 134 T − 135 , H(ζ 12 ) = −418 × T 6 − 135 .
(15)
Let η be a 13-th root of unity other than 1 and for k = 0, · · · , 12, let ζk = ζ(ηk q).
(16)
Then [3] the symmetric functions of the ζk are σ1 = ζk = 13, k
σ2 =
ζk ζl = 13T + 7 × 13,
k
σ3 =
ζk ζl ζm = 132 T + 3 × 132 ,
k
σ4 = 6 × 13T 2 + 7 × 132 T + 15 × 132 , σ5 = 74 × 13T 2 + 37 × 132 T + 5 × 133 , σ6 = 20 × 13T 3 + 38 × 132 T 2 + 134 T + 19 × 133 , σ7 = 222 × 13T 3 + 184 × 132 T 2 + 51 × 133 T + 5 × 134 , σ8 = 38 × 13T 4 + 102 × 132 T 3 + 56 × 133 T 2 + 135 T + 15 × 134 , σ9 = 346 × 13T 4 + 422 × 132 T 3 + 184 × 133 T 2 + 37 × 134 T + 3 × 135 , σ10 = 36 × 13T 5 + 126 × 132 T 4 + 102 × 133 T 3 + 38 × 134 T 2 + 7 × 135 T + 7 × 135 , σ11 = 204 × 13T 5 + 346 × 132 T 4 + 222 × 133 T 3 + 74 × 134 T 2 + 136 T + 136 σ12 = 11 × 13T 6 + 36 × 132 T 5 + 38 × 133 T 4 + 20 × 134 T 3 + 6 × 135 T 2 + 136 T + 136 and σ13 =
k
ζk =
E(q13 )14 q91 E(q169 )14 4
= T 7.
(17)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
We now determine the generating function i≥1
We have
H
i≥1
H
1 i x. ζi
⎛ ⎞ 1 i 1 ⎜⎜⎜ 1 1 1 ⎟⎟⎟ i ⎜ ⎟x x = + + · · · + ⎝ i ⎠ ζi 13 ζ0i ζ1i ζ12 i≥1 x x 1 x + + ··· + = 13 ζ0 − x ζ1 − x ζ12 − x N = , D
5
(18)
(19)
where D = (ζ0 − x)(ζ1 − x) · · · (ζ12 − x) = σ13 − σ12 x + σ11 x2 − + · · · + σ1 x12 − x13 = T 7 − (11 × 13T 6 + 36 × 132 T 5 + · · · + 136 )x + − · · · + 13x12 − x13
(20)
and x (ζl − x) 13 k=0 lk 12
N=
x ∂D 13 ∂x = (11T 6 + 36 × 13T 5 + · · · + 135 )x − + · · · − 12x12 + x13 .
=−
If we divide numerator and denominator by T 7 , we find 1 11 135 x − + · · · + 7 x13 + · · · + T T7 T 1 i H i x = . ζ 136 11 × 13 1 i≥1 + · · · + 7 x + − · · · − 7 x13 1− T T T It follows from this that H
mi, j 1 , = ζi Tj j≥1
(21)
(22)
(23)
where mi, j = 0 f or j > 7i
(24)
m2i, j = 0 f or j < i and m2i+1, j = 0 f or j < i.
(25)
and
Indeed, we have
135 1 11 + · · · + 7 x − + · · · + 7 x13 mi, j T T T xi = . j T 136 11 × 13 1 i≥1 j≥1 + · · · + 7 x + − · · · − 7 x13 1− T T T 5
(26)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
By the simple expedient of replacing
6
1 by y in (26), we obtain (12). T
From (23)–(25), we obtain 1 11 36 × 13 38 × 132 20 × 133 6 × 134 135 135 H = + + + + + 6 + 7, ζ T T2 T3 T4 T T5 T
(27)
and, for i ≥ 1, H
14i 13i 13i mi, j m2i,i+ j ai, j 1 = = = j i+ j 2i T T T i+ j ζ j=i+1 j=1 j=1
(28)
and H
1
=
ζ 2i+1
14i+7
m2i+1,i+ j 13i+7 bi, j m2i+1, j 13i+7 = = , j i+ j T T T i+ j j=i+1 j=1 j=1
(29)
where the ai, j and bi, j satisfy (11). Now, (27) becomes 7 39 169 6 q E(q169 ) q26 E(q169 )4 ) q13 E(q169 )2 2 q E(q H + 36 × 13 + 38 × 13 = 11 13 2 13 4 13 6 E(q) E(q ) E(q ) E(q ) +20 × 134
65 169 10 78 169 12 91 169 14 q52 E(q169 )8 ) ) ) 4 q E(q 5 q E(q 5 q E(q + 6 × 13 + 13 + 13 , E(q13 )8 E(q13 )10 E(q13 )12 E(q13 )14
(30)
and this reduces easily to (5). (28) becomes ⎛ 13 i+ j ⎞ 13i ⎜⎜⎜ q7 E(q169 ) 2i ⎟⎟⎟ q E(q169 )2 ⎟⎟⎠ = ⎜ ai, j , H ⎜⎝ E(q) E(q13 )2 j=1
(31)
or,
13 2i−1
13i
E(q169 )2 j , E(q13 )2 j+1
(32)
⎛ 13 i+ j ⎞ ⎜⎜⎜ q7 E(q169 ) 2i+1 ⎟⎟⎟ 13i+7 q E(q169 )2 ⎟⎟⎠ = ⎜ bi, j , H ⎜⎝ E(q) E(q13 )2 j=1
(33)
) E(q)2i
i−13 E(q
H q
=
j=1
ai, j q13 j−13
while (29) becomes
or, i−6
H q
13i+7 169 2 j−1 E(q13 )2i ) 13 j−13 E(q b q . = i, j 13 )2 j E(q)2i+1 E(q j=1
(34)
Now we prove (6) and (7) by induction. We have seen that (5) holds. This is the case α = 0 of (6). Suppose (6) holds for some α ≥ 0. Then (132α+2 −1)/24 E(q13 )2i−1 11 × 132α+1 + 1 n−12 2α+1 q p 13 n+ = x2α+1,i qi−13 . 24 E(q)2i n≥0 i=1 6
(35)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
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If we apply H to (35), we find
11 × 132α+1 + 1 13n p 132α+1 (13n + 12) + q 24 n≥0 (132α+2 −1)/24
13i
E(q169 )2 j E(q13 )2 j+1 i=1 j=1 ⎛ ⎞ (132α+3 −13)/24 ⎜(132α+2 −1)/24 ⎟⎟⎟ ⎜⎜⎜ E(q169 )2 j ⎜⎜⎜ = x2α+1,i ai, j ⎟⎟⎟⎟⎠ q13 j−13 ⎝ E(q13 )2 j+1 j=1 i=1 =
=
x2α+1,i
(132α+3 −13)/24
ai, j q13 j−13
x2α+2, j q13 j−13
j=1
E(q169 )2 j , E(q13 )2 j+1
(36)
or, (132α+3 −13)/24 E(q13 )2 j 23 × 132α+2 + 1 n 2α+2 q = p 13 n+ x2α+2, j q j−1 , 24 E(q)2 j+1 n≥0 j=1
(37)
which is (7). Now suppose (7) holds for some α ≥ 0. Then (132α+3 −13)/24 23 × 132α+2 + 1 n−5 E(q13 )2i 2α+2 p 13 n+ x2α+2,i qi−6 . q = 24 E(q)2i+1 n≥0 i=1
(38)
If we apply H to (38), we find
23 × 132α+2 + 1 13n q p 132α+2 (13n + 5) + 24 n≥0 (132α+3 −13)/24
13i+7
E(q169 )2 j−1 E(q13 )2 j i=1 j=1 ⎛ ⎞ (132α+4 −1)/24 ⎜(132α+3 −13)/24 ⎟⎟⎟ 169 2 j−1 ⎜⎜⎜ ⎜⎜⎜ ⎟⎟⎟⎟ q13 j−13 E(q ) = x b 2α+2,i i, j ⎝ ⎠ E(q13 )2 j j=1 i=1 =
=
(132α+4 −1)/24
x2α+2,i
bi, j q13 j−13
x2α+3, j q13 j−13
j=1
E(q169 )2 j−1 , E(q13 )2 j
(39)
or, (132α+4 −1)/24 11 × 132α+3 + 1 n E(q13 )2 j−1 q = p 132α+3 n + x2α+3, j q j−1 , 24 E(q)2 j n≥0 j=1
(40)
which is (6) with α + 1 for α. This completes our proof of (6) and (7). 4. The second identity in the family
n≥0
p(169n + 162)qn =
91 i=1
7
x2,i qi−1
E(q13 )2i E(q)2i+1
(41)
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
where x2 = (x2,i )i≥1 = (129913904637, 6061605043150208 × 13, 13608112586256178046 × 132 6280902061708574674384 × 133 , 80635921883086425364027 × 135 , 84554753069155774658709629 × 135 , 3916468033999952967228746399 × 136 , 116548389829851820637604548142 × 137 , 2408891668264357573695348194716 × 138 , 36596898089679538239041805817046 × 139 , 426530964811783728667060279961675 × 1310 , 3942226951585332362350072435077528 × 1311 , 385658958461803612492956251132861773 × 1311 , 185683085939890676806684250771459434 × 1313 , 983797065112634408871060753747525360 × 1314 , 4477389554472709218367190024998232130 × 1315 , 17721739170007369591672941408346874421 × 1316 , 4742355490178921142892419097596618355 × 1318 , 190225453599141520922788551641691242508 × 1318 , 524721368801814092141453950047818477326 × 1319 , 1302911978265864071115800136841270884325 × 1320 , 2929975138807498636693119222491450961566 × 1321 , 5999389868465532788358368350259321205163 × 1322 , 11238726553479300068613150748539443879990 × 1323 , 19343912101567621453437589381740398385615 × 1324 , 399199887453496794280385245259602986657786 × 1324 , 45114569049056244774820332010258592196403 × 1326 , 61531718773137637822015907451354660369102 × 1327 , 78128730639946286724152148516203955404966 × 1328 , 92588134761861713785091033497567560642309 × 1329 , 7895718682337867643451727088301900325822 × 1331 , 106674774717034569765482744541776413982076 × 1331 , 104127690533250094738334370281080119752897 × 1332 , 95632615339178385499866474240356595035734 × 1333 , 82769860945855840879428760727969455676949 × 1334 , 67607586521876860960757233146365406440864 × 1335 , 52185630770144277441328235524086183756249 × 1336 , 38112125284602081354120996414789327371490 × 1337 , 342730657709309329959063447557419449170283 × 1337 , 17291161864563075008390801545579770606804 × 1339 , 10762180018683300577592515849620716685170 × 1340 , 6361993276065687616569518872990897763233 × 1341 , 3574560685274501116380983329127804315499 × 1342 , 146936993657441362564501166152594688776 × 1344 , 8
8
M. D. Hirschhorn / Journal of Number Theory 00 (2017) 1–9
9
971409646247113487173405170610660948070 × 1344 , 470358969516921694951165237588782471677 × 1345 , 216944721415349512912594980889327147413 × 1346 , 95351253212840170160691532583542843424 × 1347 , 39948335504280947062152709269989049847 × 1348 , 15957914357030470689632293966446907145 × 1349 , 6079141248603201845764987330492593982 × 1350 , 28714246146489509001585060775229857512 × 1350 , 765494941301775657755760752212309475 × 1352 , 253052852451392618013575739199077679 × 1353 , 79788127925215402822125633710674827 × 1354 , 23992399482226475885195013027244418 × 1355 , 529171510382781913451554617989308 × 1357 , 1880313225394592406581074643938328 × 1357 , 489788423819708495595161861618881 × 1358 , 121536611116247430071845283921984 × 1359 , 28716099109510015333499605348975 × 1360 , 6456958185273171048474896073300 × 1361 , 1380835433937604032219951482224 × 1362 , 280643142907342084675152233134 × 1363 , 704129708359731486328469591897 × 1363 , 762886312702089600644337633 × 1366 , 1721000064243705676349614170 × 1366 , 282702871665192373362871298 × 1367 , 43901108707026096873874183 × 1368 , 6435317579777653828753163 × 1369 , 888969653957555557187503 × 1370 , 115507086031646435494256 × 1371 , 14086827856186927900879 × 1372 , 1608638921698649370919 × 1373 , 171538848960796539961 × 1374 , 17028556255537068094 × 1375 , 20384714589175433972 × 1375 , 1734110498546299875 × 1376 , 135634467981532940 × 1377 , 9699744171620664 × 1378 , 630048041950131 × 1379 , 36877266904170 × 1380 , 11397690381 × 1383 , 88708802688 × 1382 , 3547018629 × 1383 , 120668870 × 1384 , 3396925 × 1385 , 75992 × 1386 , 1267 × 1387 , 14 × 1388 , 1388 , 0, · · ·). (42) [1] A.O.L. Atkin and J. N. O’Brien, Some properties of p(n) and c(n) modulo powers of 13, Trans. Amer. Math. Soc. 126 (1967), 442–459. [2] G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson, eds. Ramanujan’s Collected Papers, AMS Chelsea 2000. [3] M.D. Hirschhorn, Ramanujan’s tau function, Proceedings of Alladi 60, A conference held in honor of Krishna Alladi on the occasion of his 60th Birthday, Springer, to appear. [4] S. Ramanujan, Some properties of p(n), The number of partitions of n, Proc. Camb. Philos. Soc. XIX (1919), 207–210. [5] H. S. Zuckerman, Identities analogous to Ramanujan’s identities involving the partition function, Duke Math. J. 5 (1) (1939), 88–110.
9