Remarks on a Bailey pair with one free parameter
We offer a more general Bailey pair than the pair obtained in the papers [Andrews G. E., Adv. Combin., Waterloo Workshop in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] and [Patkowski A. E., Discrete Math., 310, 961 – 965 (2010)] by two different methods.
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| author | Patkowski, A. E. Патковски, А. Е. |
| author_facet | Patkowski, A. E. Патковски, А. Е. |
| author_sort | Patkowski, A. E. |
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| description | We offer a more general Bailey pair than the pair obtained in the papers [Andrews G. E., Adv. Combin., Waterloo Workshop
in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] and [Patkowski A. E., Discrete Math., 310, 961 – 965
(2010)] by two different methods. |
| first_indexed | 2026-03-24T02:10:42Z |
| format | Article |
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UDC 517.5
A. E. Patkowski (USA)
REMARKS ON A BAILEY PAIR WITH ONE FREE PARAMETER
ЗАУВАЖЕННЯ ЩОДО ПАРИ БЕЙЛI З ОДНИМ ВIЛЬНИМ ПАРАМЕТРОМ
We offer a more general Bailey pair than the pair obtained in the papers [Andrews G. E., Adv. Combin., Waterloo Workshop
in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] and [Patkowski A. E., Discrete Math., 310, 961 – 965
(2010)] by two different methods.
Запропоновано бiльш загальну пару Бейлi, нiж та, що була отримана в роботах [Andrews G. E., Adv. Combin.,
Waterloo Workshop in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] та [Patkowski A. E., Discrete
Math., 310, 961 – 965 (2010)] двома рiзними методами.
1. Introduction. Recall that a Bailey pair (\alpha n(a, q), \beta n(a, q)) = (\alpha n, \beta n) is a pair of sequences that
satisfy (relative to a) [6]
\beta n =
\sum
i\geq 0
\alpha i
(q)n - i(aq)n+i
.
(Refer to [10] for q-series notation.) In [22] (Lemma 2.2) we offered a new Bailey pair for
\beta n =
( - 1)nqn
2
( - a)2n(q2; q2)n
, (1.1)
where a = 1, or a = q. As a direct consequence of this pair, we were able to offer new information
on the distinct rank parity function
\sigma (q) =
\sum
n\geq 0
qn(n+1)/2
(1 + q)(1 + q2) . . . (1 + qn)
.
The function \sigma (q) was shown in [1] to be related to the arithmetic of \BbbQ (
\surd
6), and therefore \BbbQ (
\surd
2),
and \BbbQ (
\surd
3). The key to this observation was the use of Bailey pairs to relate \sigma (q) to indefinite
quadratic forms.
Andrews [5, p.72] (equation (45)) also offered a proof of the pair with the \beta n given in (1.1)
with a = q using a recurrence approach, and subsequently establishing new partition theorems using
different limiting cases of Bailey’s lemma than used in [12].
2. The Bailey pair. We will apply the same proof offered in [12] but in greater generality.
Theorem 2.1. With free parameter x, we have the Bailey pair (\alpha \prime
n(a
2, x, q2), \beta \prime
n(a
2, x, q2)),
where
\alpha \prime
n(a
2, x, q2) =
qn
2 - n( - x)n(1 - a2q4n)(a2/x; q2)n
(1 - a2)(q2x; q2)n
\sum
0\leq j\leq n
(1 - aq2j - 1)(a)j - 1(x; q
2)j
qj(j - 1)/2xj(q)j(a2/x; q2)j
,
\beta \prime
n(a
2, x, q2) =
( - 1)nqn
2
(1 - x)
( - a)2n(q2; q2)n(1 - xq2n)
.
c\bigcirc A. E. PATKOWSKI, 2017
268 ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
REMARKS ON A BAILEY PAIR WITH ONE FREE PARAMETER 269
Proof. From [2] (Theorem 2.3) we have the pair (\alpha n, \beta n) relative to a, where
\alpha n(a, b, c, q) =
qn
2
(bc)n(1 - aq2n)(a/b)n(a/c)n
(1 - a)(bq, cq)n
\sum
0\leq j\leq n
( - 1)j(1 - aq2j - 1)(a)j - 1(b, c)j
qj(j - 1)/2(bc)j(q, a/b, a/c)j
, (2.1)
\beta n(a, b, c, q) =
1
(bq, cq)n
. (2.2)
Recall from [12] (equations (2.10), (2.11)) the result,
\=\alpha n(a
2, q2) =
(1 + aq2n)
(1 + a)qn
\alpha n(a, q), (2.3)
\=\beta n(a
2, q2) =
q - n
( - a; q)2n
\sum
k\geq 0
( - 1)n - kq(n - k)2 - (n - k)
(q2; q2)n - k
\beta k(a, q). (2.4)
The equations (2.3), (2.4) allow us to change the base of a Bailey pair from q to q2.
From Fine [9, p. 17] (equation (15.51)) we have\sum
i\geq 0
( - 1)n - iq\{ (n - i)2 - (n - i)\} /2
(q)n - i(bq)i
=
( - 1)nqn(n+1)/2(1 - b)
(q)n(1 - bqn)
. (2.5)
Now putting b = - c in (2.1), (2.2), replacing c by
\surd
x, and then inserting the resulting pair in (2.3),
(2.4) gives Theorem 2.1 after noting (2.5).
We now offer some corollaries as special cases of Theorem 2.1 that will be noted in the next
section.
Corollary 2.1. We have the Bailey pair
\alpha \prime
n(q
2, - q, q2) =
( - 1)nqn(n - 1)/2(1 - q2n+1)
(1 - q)
,
\beta \prime
n(q
2, - q, q2) =
( - 1)nqn
2
( - q)2n+1(q2; q2)n
.
Corollary 2.2. We have the Bailey pair
\alpha \prime
n(q
4, q, q2) =
( - 1)nqn
2
(1 - q4n+4)
(1 - q4)
\sum
0\leq j\leq n
q - j(j+1)/2,
\beta \prime
n(q
4, q, q2) =
( - 1)nqn
2
( - q)2n(q2; q2)n(1 - q4n+2)
.
Corollary 2.3. We have the Bailey pair
\alpha \prime
n(q
2, q, q2) =
( - 1)nqn
2
(1 + q2n+1)
(1 - q2)
\left( \sum
0\leq j\leq n
q - j(j+1)/2 +
\sum
0\leq j\leq n - 1
q - j(j+1)/2
\right) ,
\beta \prime
n(q
2, q, q2) =
( - 1)nqn
2
( - q)2n(q2; q2)n(1 - q2n+1)
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
270 A. E. PATKOWSKI
3. Some partitions and \bfitq -series. We will use some special instances of Bailey’s lemma [6]\sum
n\geq 0
(X)n(Y )n(aq/XY )n\beta n =
(aq/X)\infty (aq/Y )\infty
(aq)\infty (aq/XY )\infty
\sum
n\geq 0
(X)n(Y )n(aq/XY )n\alpha n
(aq/X)n(aq/Y )n
. (3.1)
In [1] Andrews et al. considered the function
\sigma \ast (q) =
\sum
n\geq 1
( - 1)nqn
2
(q; q2)n
,
which generates O(n), the number of partitions of n into odd parts with the property, that if a
number appears, then all smaller numbers appear as parts as well, weighted by - 1, if the largest part
is congruent to 1 (\mathrm{m}\mathrm{o}\mathrm{d} 4), and +1, if the largest part is congruent to 3 (\mathrm{m}\mathrm{o}\mathrm{d} 4). It was noted in [1]
that \sigma \ast (q) is also related to \BbbQ (
\surd
6). Further notes on \sigma \ast (q) can be found in [11], and more examples
related to real quadratic fields are given in [5, 7, 8, 11, 12]. We consider a similar function\sum
n\geq 1
( - 1)nqn
2
(q; q2)n(1 + q2n - 1)
=
\sum
n\geq 1
( - 1)nq1+3+...+2n - 1
(1 - q)(1 - q3) . . . (1 - q2n - 3)(1 - q(2n - 1)+(2n - 1))
. (3.2)
The q-series in (3.2) generates the same partitions counted by O(n) (and same weight function)
with the additional condition that the largest part appears an odd number of times. We denote such
partitions to be O\ast (n).
Corollary 3.1. We have
2
\sum
n\geq 1
O\ast (n)( - q)n =
\sum
n\geq 1
qn
2
n - 1\sum
j= - n
q - j(j+1)/2.
Proof. Take the Bailey pair in Corollary 2.2 and insert it into (3.1) (with a = q2, then q \rightarrow q2)
with X = q2, Y = - q2. The proof is complete after multiplying both sides by 2 upon noting that
2
\sum
0\leq j\leq n
q - j(j+1)/2 =
\sum n
j= - n - 1
q - j(j+1)/2 and then shift the resulting indefinite quadratic sum
over n with n \rightarrow n - 1.
Set L = \BbbQ (
\surd
2), let OL be the ring of integers of L, and let \u a \subset OL denote an ideal. For such
an ideal we denote its norm function to be N(\u a). Corollary 3.1 may be used to relate O\ast (n) to the
number of inequivalent elements of OL with norm N(\u a) = 2x2 - y2.
Corollary 3.2. We have
2q - 1
\sum
n\geq 1
O\ast (n)q8n =
\sum
\u a\subset OL
N(\u a)\equiv - 1 (mod 8)
( - 1)
N(\u a)+1
8 qN(\u a).
Proof. We use [1] (Lemma 3) and Corollary 3.1. Since 2( - 1)nO\ast (n) is equal to the number of
solutions of n = i2 - j(j+1)/2 with - i \leq j \leq i - 1, i \geq 1, we may write 2(2i)2 - (2j+1)2 = 8n - 1.
Any solution of N(\u a) = 2x2 - y2 = 8n - 1, n \in \BbbN , must have y odd, and subsequently x even.
Further, if we write x = 2i, y = 2j + 1, we have
- (2i) < (2j + 1) \leq (2i), i > 0. (3.3)
The solutions of (3.3) are precisely the pairs (j, i) that satisfy - i \leq j \leq i - 1, i \geq 1.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
REMARKS ON A BAILEY PAIR WITH ONE FREE PARAMETER 271
Corollary 3.3 ([3], Entry 9.4.3). We have
\sum
n\geq 0
qn(2n+1)
( - q)2n+1
=
\sum
n\geq 0
qn(3n+1)/2(1 - q2n+1).
Proof. Insert the Bailey pair in Corollary 2.1 into (3.1) with X = q2, Y \rightarrow \infty .
We mention this result in passing only to emphasize that Theorem 2.1 contains a broad range of
identities. Corollary 3.3 has appeared in [3, p. 233] (equation (9.4.4)), and was noted in [12] due to
its relevance to the q-series \sum
n\geq 0
qn(2n+1)
( - q)2n
,
which was found to be lacunary and related to \sigma (q) in [12], by using the x \rightarrow 0 instance of
Theorem 2.1.
In [7] Bringmann and Kane consider the q-series
f1(q) =
\sum
n\geq 0
qn(n+1)/2
( - q)n(1 - q2n+1)
,
and related it to the arithmetic of \BbbQ (
\surd
2). We offer some further information on f1(q).
Corollary 3.4. We have
\sum
n\geq 0
qn(n+1)
( - q2; q2)n(1 - q2n+1)
=
\sum
n\geq 0
qn(n+1)(1 + q2n+2)
n\sum
j=0
q - j(j+1)/2. (3.4)
Therefore, the “even function" of f1(q2) is a generating function for a lacunary sequence.
Proof. Let f \prime
1(q) be the left-hand side of (3.4). Then clearly, (f \prime
1(q)+ f \prime
1( - q))/2 = f1(q
2). The
result follows after inserting the Bailey pair in Corollary 2.2 into the X = q2, Y = - q3 instance of
(3.1) to get (3.4).
Another proof may be obtained using Corollary 2.3, and we leave this to the reader.
Corollary 3.5. Let s+t(n) be the number of representations of n as a sum of a triangular number
i and a square j weighted by ( - 1)j or
s+t(n) =
\sum
r\in \BbbZ ,k\geq 0
n=r2+k(k+1)/2
( - 1)r
2
.
Then \sum
n\in \BbbZ ,m\geq 0
( - 1)nqn
2+m(m+1)/2 =
\sum
n\geq 0
s+t(n)q
n =
=
\sum
n\geq 0
( - 1)nqn(n+1)
\left( \sum
0\leq j\leq n
q - j(j+1)/2 +
\sum
0\leq j\leq n - 1
q - j(j+1)/2
\right) .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
272 A. E. PATKOWSKI
Proof. We use the X = - q, Y = - q2 case of (3.1) coupled with the Bailey pair in Corollary
2.3 and invoke the identity (a special limiting case of [9, p. 18] (equation (16.3))
\sum
n\geq 0
( - 1)nqn(n+1)(1 - x)
(q2; q2)n(1 - xq2n)
=
(q2; q2)\infty
(xq2; q2)\infty
.
We note in closing that Corollary 3.5 gives a mapping between the number of inequivalent
elements of OL with norm 8n + 1 to the number of inequivalent elements of OL\prime , where L\prime =
= \BbbQ (
\surd
- 2), with norm 8n+ 1.
We also mention it is possible to obtain a more general expansion for the product
(q)\infty (q2; q2)\infty
( - q)\infty (xq2; q2)\infty
=
\sum
n\geq 0
qn
2
( - x)n(1 - q2n+1)
(q2/x; q2)n
(q2x; q2)n
\sum
0\leq j\leq n
(1 + qj)(x; q2)j
qj(j - 1)/2xj(q2/x; q2)j
, (3.5)
using the same limiting case of (3.1) with the a = q case of Theorem 2.1. A nice corollary of (3.5)
with x \rightarrow 0 is the famous expansion due to Rogers [13] for the weight 1 modular form\prod
n\geq 1
(1 - qn)2 =
\sum
n\geq 0
qn(2n+1)(1 - q2n+1)
\sum
| j| \leq n
( - 1)jq - j(3j+1)/2.
References
1. Andrews G. E., Dyson F. J., Hickerson D. Partitions and indefinite quadratic forms // Invent. Math. – 1988. – 91. –
P. 391 – 407.
2. Andrews G. E., Hickerson D. Ramanujan’s “lost" notebook. VII. The sixth order mock theta functions // Adv. Math. –
1991. – 89, № 1. – P. 60 – 105.
3. Andrews G. E., Berndt B. C. Ramanujan’s lost notebook. Part I. – New York: Springer, 2005.
4. Andrews G. E. Partitions with distinct evens // Adv. Combin. Math. Proc. Waterloo Workshop in Comput. Algebra,
2008 / Eds I. Kotsireas and E. Zima. – Springer, 2009. – P. 31 – 37.
5. Andrews G. E. Partitions with early conditions // Adv. Combin., Waterloo Workshop in Comput. Algebra, May
26 – 29, 2011 / Eds I. S. Kotsireas and E. V. Zima. – Springer, 2013. – P. 57 – 76
6. Bailey W. N. Identities of the Rogers – Ramanujan type // Proc. London Math. Soc. – 1949. – 50, № 2. – P. 1 – 10.
7. Bringmann K., Kane B. Multiplicative q-hypergeometric series arising from real quadratic fields // Trans. Amer.
Math. Soc. – 2011. – 363, № 4. – P. 2191 – 2209.
8. Corson D., Favero D., Liesinger K., Zubairy S. Characters and q-series in \BbbQ (
\surd
2) // J. Number Theory. – 2004. –
107. – P. 392 – 405.
9. Fine N. J. Basic hypergeometric series and applications // Math. Surv. – Providence: Amer. Math. Soc.,
1988. – 27.
10. Gasper G., Rahman M. Basic hypergeometric series. – Cambridge: Cambridge Univ. Press, 1990.
11. Patkowski A. On some partition functions related to some mock Theta functions // Bull. Pol. Acad. Sci. Math. – 2009. –
57. – P. 1 – 8.
12. Patkowski A. A note on the rank parity function // Discrete Math. – 2010. – 310. – P. 961 – 965.
13. Rogers L. J. Second memoir on the expansion of certain infinite products // Proc. London Math. Soc. – 1894. – 25. –
P. 318 – 343.
Received 13.10.15
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 2
|
| id | umjimathkievua-article-1692 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:10:42Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/8f/c30d335821c58aa10e7ba2b993499e8f.pdf |
| spelling | umjimathkievua-article-16922019-12-05T09:23:56Z Remarks on a Bailey pair with one free parameter Зауваження щодо пари бейлi з одним вiльним параметром Patkowski, A. E. Патковски, А. Е. We offer a more general Bailey pair than the pair obtained in the papers [Andrews G. E., Adv. Combin., Waterloo Workshop in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] and [Patkowski A. E., Discrete Math., 310, 961 – 965 (2010)] by two different methods. Запропоновано бiльш загальну пару Бейлi, нiж та, що була отримана в роботах [Andrews G. E., Adv. Combin., Waterloo Workshop in Comput. Algebra (May 26 – 29, 2011), Springer (2013), p. 57 – 76] та [Patkowski A. E., Discrete Math., 310, 961 – 965 (2010)] двома рiзними методами. Institute of Mathematics, NAS of Ukraine 2017-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1692 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 2 (2017); 268-272 Український математичний журнал; Том 69 № 2 (2017); 268-272 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1692/674 Copyright (c) 2017 Patkowski A. E. |
| spellingShingle | Patkowski, A. E. Патковски, А. Е. Remarks on a Bailey pair with one free parameter |
| title | Remarks on a Bailey pair with one free parameter |
| title_alt | Зауваження щодо пари бейлi з одним вiльним параметром |
| title_full | Remarks on a Bailey pair with one free parameter |
| title_fullStr | Remarks on a Bailey pair with one free parameter |
| title_full_unstemmed | Remarks on a Bailey pair with one free parameter |
| title_short | Remarks on a Bailey pair with one free parameter |
| title_sort | remarks on a bailey pair with one free parameter |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1692 |
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