Proofs of some conjectures of Chan on Appell–Lerch sums
On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum ϕ ( q ) : = ∑ n = 0 ∞ ( - q ; q ) 2 n q n + 1 ( q ; q 2 ) n + 1 2 , which is connected to some of his sixth order mock theta functions. Let ∑ n = 1 ∞ a ( n ) q n : = ϕ ( q ) . In this paper, we find a representation of the genera...
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| Published in | The Ramanujan journal Vol. 51; no. 1; pp. 99 - 115 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
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| Abstract | On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum
ϕ
(
q
)
:
=
∑
n
=
0
∞
(
-
q
;
q
)
2
n
q
n
+
1
(
q
;
q
2
)
n
+
1
2
,
which is connected to some of his sixth order mock theta functions. Let
∑
n
=
1
∞
a
(
n
)
q
n
:
=
ϕ
(
q
)
. In this paper, we find a representation of the generating function of
a
(
10
n
+
9
)
in terms of
q
-products. As corollaries, we deduce the congruences
a
(
50
n
+
19
)
≡
a
(
50
n
+
39
)
≡
a
(
50
n
+
49
)
≡
0
(
mod
25
)
as well as
a
(
1250
n
+
250
r
+
219
)
≡
0
(
mod
125
)
, where
r
=
1
, 3, and 4. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell–Lerch sums. |
|---|---|
| AbstractList | On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum ϕ(q):=∑n=0∞(-q;q)2nqn+1(q;q2)n+12,which is connected to some of his sixth order mock theta functions. Let ∑n=1∞a(n)qn:=ϕ(q). In this paper, we find a representation of the generating function of a(10n+9) in terms of q-products. As corollaries, we deduce the congruences a(50n+19)≡a(50n+39)≡a(50n+49)≡0(mod25) as well as a(1250n+250r+219)≡0(mod125), where r=1, 3, and 4. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell–Lerch sums. On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum ϕ ( q ) : = ∑ n = 0 ∞ ( - q ; q ) 2 n q n + 1 ( q ; q 2 ) n + 1 2 , which is connected to some of his sixth order mock theta functions. Let ∑ n = 1 ∞ a ( n ) q n : = ϕ ( q ) . In this paper, we find a representation of the generating function of a ( 10 n + 9 ) in terms of q -products. As corollaries, we deduce the congruences a ( 50 n + 19 ) ≡ a ( 50 n + 39 ) ≡ a ( 50 n + 49 ) ≡ 0 ( mod 25 ) as well as a ( 1250 n + 250 r + 219 ) ≡ 0 ( mod 125 ) , where r = 1 , 3, and 4. The first three congruences were conjectured by Chan in 2012, whereas the congruences modulo 125 are new. We also prove two more conjectural congruences of Chan for the coefficients of two Appell–Lerch sums. |
| Author | Baruah, Nayandeep Deka Begum, Nilufar Mana |
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| Cites_doi | 10.1016/j.aim.2014.07.018 10.1090/S0002-9939-2010-10538-5 10.1142/S1793042112500066 10.1007/s11005-005-0039-1 10.1007/978-1-4612-0965-2 10.24033/asens.236 10.1090/stml/034 10.1016/0001-8708(91)90083-J 10.1007/s11139-012-9379-5 10.1016/j.jmaa.2017.11.035 10.1112/plms/pdu007 10.24033/asens.248 10.1093/qjmath/53.2.147 10.1142/S1793042118501191 10.24033/asens.272 10.1007/s00208-016-1390-5 10.4064/aa153-2-3 |
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| Keywords | Mock theta function Theta function Congruence Primary 11P83 Appell–Lerch sum Secondary 33D15 |
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| References | Berndt (CR7) 1991 Appell (CR4) 1886; 3 Qu, Wang, Yao (CR20) 2018; 460 Mortenson (CR16) 2012; 29 Mortenson (CR17) 2014; 264 Berndt (CR6) 2006 Choi (CR10) 2002; 53 Appell (CR3) 1885; 2 Ramanujan (CR18) 1919; 19 Hikami (CR14) 2006; 75 Lerch (CR15) 1892; 24 Baruah, Begum (CR5) 2018; 14 CR11 CR22 Hickerson, Mortenson (CR13) 2017; 367 Chan, Mao (CR9) 2012; 8 Andrews, Hickerson (CR1) 1991; 89 Hickerson, Mortenson (CR12) 2014; 109 Ramanujan (CR19) 1988 Chan (CR8) 2012; 153 Appell (CR2) 1884; 1 Waldherr (CR21) 2011; 139 P Appell (76_CR3) 1885; 2 S Ramanujan (76_CR19) 1988 DR Hickerson (76_CR12) 2014; 109 BC Berndt (76_CR6) 2006 Y-S Choi (76_CR10) 2002; 53 S Ramanujan (76_CR18) 1919; 19 GE Andrews (76_CR1) 1991; 89 SH Chan (76_CR8) 2012; 153 SH Chan (76_CR9) 2012; 8 P Appell (76_CR4) 1886; 3 ND Baruah (76_CR5) 2018; 14 M Lerch (76_CR15) 1892; 24 ET Mortenson (76_CR17) 2014; 264 ET Mortenson (76_CR16) 2012; 29 BC Berndt (76_CR7) 1991 M Waldherr (76_CR21) 2011; 139 76_CR11 76_CR22 P Appell (76_CR2) 1884; 1 DR Hickerson (76_CR13) 2017; 367 YK Qu (76_CR20) 2018; 460 K Hikami (76_CR14) 2006; 75 |
| References_xml | – ident: CR22 – volume: 264 start-page: 236 year: 2014 end-page: 260 ident: CR17 article-title: On the dual nature of partial theta functions and Appell-Lerch sums publication-title: Adv. Math. doi: 10.1016/j.aim.2014.07.018 contributor: fullname: Mortenson – volume: 139 start-page: 865 year: 2011 end-page: 879 ident: CR21 article-title: On certain explicit congruences for mock theta functions publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2010-10538-5 contributor: fullname: Waldherr – volume: 8 start-page: 111 year: 2012 end-page: 123 ident: CR9 article-title: Two congruences for Appell-Lerch sums publication-title: Int. J. Number Theory doi: 10.1142/S1793042112500066 contributor: fullname: Mao – volume: 75 start-page: 93 year: 2006 end-page: 98 ident: CR14 article-title: Transformation formula of the “second” order mock theta function publication-title: Lett. Math. 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Phys. doi: 10.1007/s11005-005-0039-1 contributor: fullname: K Hikami – volume: 153 start-page: 161 year: 2012 ident: 76_CR8 publication-title: Acta Arith. doi: 10.4064/aa153-2-3 contributor: fullname: SH Chan – volume: 14 start-page: 1995 year: 2018 ident: 76_CR5 publication-title: Int. J. Number Theory doi: 10.1142/S1793042118501191 contributor: fullname: ND Baruah – volume: 367 start-page: 373 year: 2017 ident: 76_CR13 publication-title: Math. Ann. doi: 10.1007/s00208-016-1390-5 contributor: fullname: DR Hickerson – volume: 109 start-page: 382 year: 2014 ident: 76_CR12 publication-title: Proc. Lond. Math. Soc. doi: 10.1112/plms/pdu007 contributor: fullname: DR Hickerson – volume-title: Ramanujan’s Notebooks, Part III year: 1991 ident: 76_CR7 doi: 10.1007/978-1-4612-0965-2 contributor: fullname: BC Berndt – volume: 460 start-page: 232 year: 2018 ident: 76_CR20 publication-title: J. Math. Anal. Appl. doi: 10.1016/j.jmaa.2017.11.035 contributor: fullname: YK Qu – volume-title: Number Theory in the Spirit of Ramanujan year: 2006 ident: 76_CR6 doi: 10.1090/stml/034 contributor: fullname: BC Berndt – volume: 89 start-page: 60 year: 1991 ident: 76_CR1 publication-title: Adv. Math. doi: 10.1016/0001-8708(91)90083-J contributor: fullname: GE Andrews – volume: 3 start-page: 9 year: 1886 ident: 76_CR4 publication-title: Ann. Sci. Éc. Norm. Supér. doi: 10.24033/asens.272 contributor: fullname: P Appell – volume: 8 start-page: 111 year: 2012 ident: 76_CR9 publication-title: Int. J. Number Theory doi: 10.1142/S1793042112500066 contributor: fullname: SH Chan – volume: 139 start-page: 865 year: 2011 ident: 76_CR21 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-2010-10538-5 contributor: fullname: M Waldherr – ident: 76_CR22 – volume-title: The Lost Notebook and Other Unpublished Papers year: 1988 ident: 76_CR19 contributor: fullname: S Ramanujan – volume: 2 start-page: 9 year: 1885 ident: 76_CR3 publication-title: Ann. Sci. Éc. Norm. Supér. doi: 10.24033/asens.248 contributor: fullname: P Appell – volume: 24 start-page: 465 year: 1892 ident: 76_CR15 publication-title: Prag. České Ak. Fr. Jos. Rozpr. contributor: fullname: M Lerch – volume: 29 start-page: 121 year: 2012 ident: 76_CR16 publication-title: Ramanujan J. doi: 10.1007/s11139-012-9379-5 contributor: fullname: ET Mortenson – volume: 264 start-page: 236 year: 2014 ident: 76_CR17 publication-title: Adv. Math. doi: 10.1016/j.aim.2014.07.018 contributor: fullname: ET Mortenson |
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| Snippet | On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum
ϕ
(
q
)
:
=
∑
n
=
0
∞
(
-
q
;
q
)
2
n
q
n
+
1
(
q
;
q
2
)
n
+
1
2
,
which is connected... On page 3 of his lost notebook, Ramanujan defines the Appell–Lerch sum ϕ(q):=∑n=0∞(-q;q)2nqn+1(q;q2)n+12,which is connected to some of his sixth order mock... |
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| SubjectTerms | Combinatorics Congruences Field Theory and Polynomials Fourier Analysis Functions of a Complex Variable Mathematics Mathematics and Statistics Number Theory Sums |
| Title | Proofs of some conjectures of Chan on Appell–Lerch sums |
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