On combinatorial extensions of some Ramanujan’s mock theta functions
Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan. Using a bijection between the associated lattic...
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| author | Goyal, M. Goyal, M. Goyal, M. |
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| description | Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan. Using a bijection between the associated lattice path functions and the $(n+t)$-color partitions and then between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial identities by using bijection between the $(n+t)$-color partitions and the partitions with certain antihook differences. These interesting results present elegant combinatorial links between Ramanujan's mock theta functions, $(n+t)$-color partitions, weighted lattice paths, associated lattice paths, and antihook differences.
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UDC 517.5
M. Goyal (IK Gujral Punjab Techn. Univ., Jalandhar, India)
ON COMBINATORIAL EXTENSIONS
OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS
ПРО КОМБIНАТОРНI ПРОДОВЖЕННЯ
ДЕЯКИХ ФIКТИВНИХ ТЕТА-ФУНКЦIЙ РАМАНУДЖАНА
Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path
functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions
of Ramanujan. Using a bijection between the associated lattice path functions and the (n + t)-color partitions and then
between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and
Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial
identities by using bijection between the (n+ t)-color partitions and the partitions with certain antihook differences. These
interesting results present elegant combinatorial links between Ramanujan’s mock theta functions, (n+ t)-color partitions,
weighted lattice paths, associated lattice paths, and antihook differences.
Наведено комбiнаторну iнтерпретацiю п’яти фiктивних тета-функцiй С. Рамануджана за допомогою деяких асо-
цiйованих ґратчастих функцiй шляху та антигачкових рiзниць. Отриманi результати дають нову комбiнаторну iн-
терпретацiю п’яти фiктивних тета-функцiй Рамануджана. За допомогою взаємно однозначної вiдповiдностi мiж
асоцiйованими ґратчастими функцiями шляху та (n + t)-кольоровими розбиттями, а також мiж асоцiйованими
ґратчастими функцiями шляху та зваженими ґратчастими функцiями шляху узагальнено роботи Агарвала та Агар-
вала i Рана на випадок п’яти нових 3-шляхових комбiнаторних тотожностей. Цi результати потiм розширено на
випадок 4-шляхових комбiнаторних тотожностей за допомогою взаємно однозначної вiдповiдностi мiж (n + t)-
кольоровими розбиттями та розбиттями з певними антигачковими рiзницями. Цi цiкавi результати встановлюють
елегантнi комбiнаторнi зв’язки мiж фiктивними тета-функцiями Рамануджана, (n + t)-кольоровими розбиттями,
зваженими ґратчастими шляхами, асоцiйованими ґратчастими шляхами та антигачковими рiзницями.
1. Introduction. The world of mathematics owe a lot to the pioneering insights of the great Indian
mathematician S. Ramanujan. The list of interesting directions pioneered by him is huge. Ramanujan
has offered many important discoveries in different fields of mathematics such as in additive number
theory, probabilistic number theory, theta functions, exponential sum (now called Ramanujan sum),
zeta values, tau functions, modular equations, elliptic functions, magic squares, generating functions,
continued fraction, Eulerian series, partitions, combinatorics and many more. His work is still being
pursued actively and fruitfully (see, for instance, [10, 17, 19, 22]). The last gift of Ramanujan to the
world of mathematics is “The Mock Theta Functions”. In particular, Ramanujan gave a list of 17
mock theta functions which he divided into three classes. He defined four 3rd order, ten 5th order
and three 7th order mock theta functions. For the definitions and orders of the mock theta functions,
the interested readers are referred to [13]. Agarwal in [4, 5] interpreted four mock theta functions
of Ramanujan combinatorially using n-color partitions and weighted lattice paths. One more mock
theta function of order five had been interpreted combinatorially by Agarwal and Rana [12] using
(n + 2)-color partitions and weighted lattice paths. The purpose of this paper is to further explore
these five mock theta functions using associated lattice paths as defined and studied by Anand and
Agarwal [14] and antihook differences as defined by Agarwal and Andrews [6].
Let us first recall some definitions:
Definition 1.1 [7]. A partition with “(n+ t) copies of n”, t \geq 0, is a partition in which a part
of size n, n \geq 0, can come in (n+ t) different colors denoted by subscripts: n1, n2, . . . , nn+t. Note
c\bigcirc M. GOYAL, 2020
46 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 47
that zeros are permitted if and only if t is greater than or equal to one. Also, zeros are not permitted
to repeat in any partition.
Remark 1.1. We note that if we take t = 0, then these are nothing but the n-color partitions.
Definition 1.2. The weighted difference of two parts gk, hl (g \geq h) is defined by g - h - k - l
and is denoted by ((gk - hl)).
In [8] the weighted lattice paths are described as:
Definition 1.3. All paths will be of finite length lying in the first quadrant. They will begin on
the Y -axis and terminate on the X-axis. Only three moves are allowed at each step:
northeast: from (x, y) to (x+ 1, y + 1);
southeast: from (x, y) to (x+ 1, y - 1), only allowed if y > 0;
horizontal: from (x, 0) to (x+ 1, 0), only allowed along X-axis.
All our lattice paths are either empty or terminate with a southeast step: from (x, 1) to (x+1, 0).
In describing lattice paths, the following terminology is used:
Peak: Either a vertex on the Y -axis which is followed by a southeast step or a vertex preceded
by a northeast step and followed by a southeast step.
Valley: A vertex preceded by a southeast step and followed by a northeast step. Note that a
southeast step followed by a horizontal step followed by a northeast step does not constitute a valley.
Mountain: A section of the path which starts on either the X- or Y -axis, which ends on the
X-axis and which does not touch the X-axis anywhere in between the end points. Every mountain
has at least one peak and may have more than one.
Plain: A section of the path consisting of only horizontal steps which starts either on the Y -axis
or at a vertex preceded by a southeast step and ends at a vertex followed by a northeast step.
Height: Height of a vertex is its y-coordinate.
Weight: Weight of a vertex is its x-coordinate.
Weight of a Lattice Path: It is the sum of the weights of its peaks.
Anand and Agarwal [14] gave the following description of associated lattice paths.
Definition 1.4. All paths will be of finite length lying in the first quadrant. They will begin on
the Y -axis and terminate on the X-axis. Only three moves are allowed at each step:
northeast: from (x, y) to (x+ 1, y + 1);
southeast: from (x, y) to (x+ 1, y - 1), only allowed if y > 0;
horizontal: from (x, y) to (x+1, y), only allowed when the first step is preceded by a northeast
step and the last is followed by a southeast step.
The following terminology is used in describing associated lattice paths:
Truncated Isosceles Trapezoidal Section (TITS): A section of the path which starts on the X-axis
with northeast steps followed by horizontal steps and then followed by southeast steps ending on the
X-axis forms a TITS.
Since the lower base lies on X-axis and is not a part of the path, hence the term truncated.
Slant Section (SS): A section of the path consisting of only southeast steps which starts on the
Y -axis (origin not included) and ends on the X-axis.
Height of a slant section: It is “t” if it starts from (0, t). Clearly, a path can have an SS only in
the beginning of the path. An associated lattice path can have at most one SS.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
48 M. GOYAL
Fig. 1. One SS of height 1 and one TITS with ordered pair \{ 2, 3\} .
Weight of a TITS: This is defined by representing every TITS by an ordered pair \{ a, b\} where a
denotes its altitude and b the length of the upper base. Weight of a TITS with ordered pair \{ a, b\} is
a units.
Weight of an Associated Lattice Path: It is the sum of weights of its TITSs.
Note that Slant Section is assigned weight zero.
Example 1.1. In this example, the associated lattice path has one SS of height 1 and one TITS
with ordered pair \{ 2, 3\} and its weight is 2 units (see Fig. 1).
Agarwal and Andrews [6] gave the following definition of antihook differences.
Definition 1.5. Let \Pi be a partition whose Ferrers graph is embedded in the fourth quadrant.
Each node (x, y) of the fourth quadrant which is not in the Ferrers graph of \Pi is said to possess
an antihook difference \xi x - \zeta y relative to \Pi , where \xi x is the number of nodes in the xth row of
the fourth quadrant to the left of node (x, y) that are not in the Ferrers graph of \Pi and \zeta y is the
number of nodes in the yth column of the fourth quadrant that lie above node (x, y) and are not in
the Ferrers graph of \Pi .
Definition 1.6. The nodes (x, y) of \Pi for which x - y = d are said to lie on diagonal d.
Definition 1.7. The rank of a partition is defined as the largest part minus the number of parts.
Definition 1.8. A right angle in the Ferrers graph of a partition is called a hook and will be
denoted by [p, q] if there are p nodes in the row and q nodes in the column. Thus, for instance, [6,
4] represents the hook
\bullet \bullet \bullet \bullet \bullet \bullet
\bullet
\bullet
\bullet
Definition 1.9 [15]. A two rowed array of nonnegative integers\Biggl(
p1 p2 . . . p\nu
q1 q2 . . . q\nu
\Biggr)
,
where p1 \geq p2 \geq . . . \geq p\nu \geq 0, q1 \geq q2 \geq . . . \geq q\nu \geq 0 is known as a generalized Frobenius
partition or more simply an F-partition of \mu if p1 + p2 + . . .+ p\nu + q1 + q2 + . . .+ q\nu + \nu = \mu .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 49
For example, \mu = 28 = 4+ (6 + 5+ 2+ 0) + (5 + 3+ 2+ 1) and the corresponding Frobenius
symbol is \Biggl(
6 5 2 0
5 3 2 1
\Biggr)
.
The corresponding Ferrers graph is
and the associated antihook differences are given by
The following are the five mock theta functions of S. Ramanujan:
\Psi (q) =
\infty \sum
n=1
qn
2
(q; q2)n
, (1.1)
F0(q) =
\infty \sum
n=0
q2n
2
(q; q2)n
, (1.2)
\Phi 0(q) =
\infty \sum
n=0
qn
2
( - q; q2)n, (1.3)
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
50 M. GOYAL
\Phi 1(q) =
\infty \sum
n=1
qn
2
( - q; q2)n - 1, (1.4)
F1(q) =
\infty \sum
n=0
q2n(n+1)
(q; q2)n+1
, (1.5)
where
(a; q)n =
\infty \prod
i=0
(1 - aqi)
(1 - aqn+i)
.
Mock theta functions (1.1) – (1.4) were interpreted by Agarwal [4, 5] in the form of following theo-
rems.
Theorem 1.1. For \mu \geq 1, let A1(\mu ) denote the number of n-color partitions of \mu such that
(i) even parts appear with even subscripts and odd with odd, (ii) for some k, kk is a part and
(iii) the weighted difference between any two consecutive parts is 0. Let B1(\mu ) denote the number
of lattice paths of weight \mu which start at (0, 0), such that (iv) they have no valley above height 0
and (v) there is no plain. Then A1(\mu ) = B1(\mu ) for all \mu and
\infty \sum
\mu =1
A1(\mu )q
\mu =
\infty \sum
\mu =1
B1(\mu )q
\mu = \Psi (q). (1.6)
Theorem 1.2. For \mu \geq 0, let A2(\mu ) denote the number of n-color partitions of \mu such that
(i) even parts appear with even subscripts and odd with odd subscripts > 1, (ii) for some k, kk is a
part and (iii) the weighted difference between any two consecutive parts is 0. Let B2(\mu ) denote the
number of lattice paths of weight \mu which start at (0, 0), such that (iv) they have no valley above
height 0, (v) there is no plain and (vi) the height of each peak is \geq 2. Then A2(\mu ) = B2(\mu ) for all
\mu and
\infty \sum
\mu =0
A2(\mu )q
\mu =
\infty \sum
\mu =0
B2(\mu )q
\mu = F0(q). (1.7)
Theorem 1.3. For \mu \geq 0, let A3(\mu ) denote the number of n-color partitions of \mu such that
(i) the parts are of the form (2j - 1)1 or (2j)2, (ii) the minimum part is 11 or 22 and (iii) the
weighted difference between any two consecutive parts is 0. Let B3(\mu ) denote the number of lattice
paths of weight \mu which start at (0, 0), such that (iv) they have no valley above height 0, (v) there
is no plain and (vi) the height of each peak of odd weight is 1 while that of even weight is 2. Then
A3(\mu ) = B3(\mu ) for all \mu and
\infty \sum
\mu =0
A3(\mu )q
\mu =
\infty \sum
\mu =0
B3(\mu )q
\mu = \Phi 0(q). (1.8)
Theorem 1.4. For \mu \geq 1, let A4(\mu ) denote the number of n-color partitions of \mu such that
(i) the parts are of the form (2j - 1)1 or (2j)2, (ii) the minimum part is 11 and (iii) the weighted
difference between any two consecutive parts is 0. Let B4(\mu ) denote the number of lattice paths of
weight \mu which start at (0, 0), such that (iv) they have no valley above height 0, (v) there is no
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 51
plain, (vi) the height of each peak of odd weight is 1 while that of even weight is 2 and (vii) the
weight of the first peak is 1. Then A4(\mu ) = B4(\mu ) for all \mu and
\infty \sum
\mu =1
A4(\mu )q
\mu =
\infty \sum
\mu =1
B4(\mu )q
\mu = \Phi 1(q). (1.9)
Mock theta function (1.5) was interpreted by Agarwal and Rana [12] in the form of following
theorem.
Theorem 1.5. For \mu \geq 0, let A5(\mu ) denote the number of (n + 2)-color partitions of \mu such
that (i) even parts appear with even subscripts and odd with odd subscripts > 1, (ii) for some i, ii+2
is a part and (iii) the weighted difference between any two consecutive parts is 0. Let B5(\mu ) denote
the number of lattice paths of weight \mu which start at (0, 2), such that (iv) they have no valley above
height 0, (v) there is no plain and (vi) the height of each peak is \geq 2. Then A5(\mu ) = B5(\mu ) for all
\mu and
\infty \sum
\mu =0
A5(\mu )q
\mu =
\infty \sum
\mu =0
B5(\mu )q
\mu = F1(q). (1.10)
In next section, we propose to further extend Theorems 1.1 – 1.5 using associated lattice paths. We
will show that certain restricted associated lattice paths are also generated by the extreme right-hand
sides of (1.6) – (1.10). This extends Theorems 1.1 – 1.5 to five new 3-way combinatorial identities.
In Section 3, using antihook differences we further extend these results to 4-way combinatorial
identities. In last section we conclude the results and pose some open problems.
2. Combinatorial interpretations by using associated lattice paths. In this section the com-
binatorial interpretations of the mock theta functions (1.1) – (1.5) are given in terms of associated
lattice paths. These results extend the work of Agarwal [4, 5] and Agarwal and Rana [12] to five
new 3-way combinatorial identities.
2.1. Main results.
Theorem 2.1. For \mu \geq 1, let C1(\mu ) denote the number of associated lattice paths of weight
\mu such that (i) for any TITS with ordered pair \{ a, b\} , b does not exceed a, (ii) the TITSs are
arranged in order of nondecreasing altitudes and the TITSs with same altitude are ordered by the
length of their upper base, (iii) there is always a TITS with ordered pair \{ a, a\} and (iv) for any
two TITSs with respective ordered pairs \{ a1, b1\} and \{ a2, b2\} (a1 \leq a2), a2 - b2 = a1 + b1. Then
A1(\mu ) = B1(\mu ) = C1(\mu ) for all \mu and
\infty \sum
\mu =1
A1(\mu )q
\mu =
\infty \sum
\mu =1
B1(\mu )q
\mu =
\infty \sum
\mu =1
C1(\mu )q
\mu = \Psi (q).
Theorem 2.2. For \mu \geq 0, let C2(\mu ) denote the number of associated lattice paths of weight \mu
such that (i) for any TITS with ordered pair \{ a, b\} , b does not exceed a, (ii) the TITSs are arranged
in order of nondecreasing altitudes and the TITSs with same altitude are ordered by the length of
their upper base, (iii) there is always a TITS with ordered pair \{ a, a\} , (iv) the length of each upper
base is greater than 1 and (v) for any two TITSs with respective ordered pairs \{ a1, b1\} and \{ a2, b2\}
(a1 \leq a2), a2 - b2 = a1 + b1. Then A2(\mu ) = B2(\mu ) = C2(\mu ) for all \mu and
\infty \sum
\mu =0
A2(\mu )q
\mu =
\infty \sum
\mu =0
B2(\mu )q
\mu =
\infty \sum
\mu =0
C2(\mu )q
\mu = F0(q).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
52 M. GOYAL
Theorem 2.3. For \mu \geq 0, let C3(\mu ) denote the number of associated lattice paths of weight \mu
such that (i) for any TITS with ordered pair \{ a, b\} , b does not exceed a, (ii) the TITSs are arranged
in order of nondecreasing altitudes and the TITSs with same altitude are ordered by the length of
their upper base, (iii) if \{ a1, b1\} is the ordered pair of the first TITS of the path, then a1 = b1 = 1
or a1 = b1 = 2, (iv) the length of the upper base of a TITS with odd weight is 1 while that of even
weight is 2 and (v) for any two TITSs with respective ordered pairs \{ a1, b1\} and \{ a2, b2\} (a1 \leq a2),
a2 - b2 = a1 + b1. Then A3(\mu ) = B3(\mu ) = C3(\mu ) for all \mu and
\infty \sum
\mu =0
A3(\mu )q
\mu =
\infty \sum
\mu =0
B3(\mu )q
\mu =
\infty \sum
\mu =0
C3(\mu )q
\mu = \Phi 0(q).
Theorem 2.4. For \mu \geq 1, Let C4(\mu ) denote the number of associated lattice paths of weight \mu
such that (i) for any TITS with ordered pair \{ a, b\} , b does not exceed a, (ii) the TITSs are arranged
in order of nondecreasing altitudes and the TITSs with same altitude are ordered by the length of their
upper base, (iii) if \{ a1, b1\} is the ordered pair of the first TITS of the path, then a1 = b1 = 1, (iv) the
length of the upper base of a TITS with odd weight is 1 while that of even weight is 2 and (v) for
any two TITSs with respective ordered pairs \{ a1, b1\} and \{ a2, b2\} (a1 \leq a2), a2 - b2 = a1 + b1.
Then A4(\mu ) = B4(\mu ) = C4(\mu ) for all \mu and
\infty \sum
\mu =1
A4(\mu )q
\mu =
\infty \sum
\mu =1
B4(\mu )q
\mu =
\infty \sum
\mu =1
C4(\mu )q
\mu = \Phi 1(q).
Theorem 2.5. For \mu \geq 0, let C5(\mu ) denote the number of associated lattice paths of weight \mu
such that (i) for any TITS with ordered pair \{ a, b\} , b does not exceed (a + 2), (ii) the TITSs are
arranged in order of nondecreasing altitudes and the TITSs with same altitude are ordered by the
length of their upper base, (iii) there is one TITS with ordered pair \{ a, a + 2\} or an SS of height
2, (iv) the length of each upper base is greater than 1 and (v) for any two TITSs with respective
ordered pairs \{ a1, b1\} and \{ a2, b2\} (a1 \leq a2), a2 - b2 = a1 + b1. Then A5(\mu ) = B5(\mu ) = C5(\mu )
for all \mu and
\infty \sum
\mu =0
A5(\mu )q
\mu =
\infty \sum
\mu =0
B5(\mu )q
\mu =
\infty \sum
\mu =0
C5(\mu )q
\mu = F1(q).
Detailed proof of Theorem 2.1 is discussed in the next subsection and the outline of the proofs
of the remaining theorems are given in Subsection 2.3.
2.2. Proof of Theorem 2.1. The proof comprises of three steps. In first step we will show that
the right-hand side of (1.1) generates the associated lattice paths enumerated by C1(\mu ). Then we will
show a bijection between the n-color partitions enumerated by A1(\mu ) and the the associated lattice
paths enumerated by C1(\mu ). Finally, we will establish a bijection between the weighted lattice paths
enumerated by B1(\mu ) and the associated lattice paths enumerated by C1(\mu ).
Step I. We shall prove that
\infty \sum
\mu =1
C1(\mu )q
\mu =
\infty \sum
m=1
qm
2
(q; q2)m
= \Psi (q). (2.1)
In
qm
2
(q; q2)m
the factor qm
2
generates an associated lattice path having m TITSs such that ith TITS
have the ordered pair \{ 2i - 1, 1\} .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 53
For m = 3, the path begins as (see Fig. 2):
Fig. 2. TITSs for m = 3.
In the Fig. 3 we consider two successive TITSs, say, ith and (i + 1)th. Their corresponding
ordered pairs are \{ 2i - 1, 1\} and \{ 2i+ 1, 1\} , respectively.
Fig. 3. ith and (i+ 1)th TITSs.
The factor
1
(q; q2)m
generates m nonnegative multiples of (2i - 1), 1 \leq i \leq m, say, \beta 1 \times
\times 1, \beta 2 \times 3, . . . , \beta m \times (2m - 1). This is encoded by increasing the altitude of ith TITS by 2(\beta m +
+ \beta m - 1 + . . .+ \beta m - i+2) + \beta m - i+1 and the length of the upper base by \beta m - i+1. So the associated
ordered pair becomes
\bigl\{
2i - 1 + 2(\beta m + \beta m - 1 + . . .+ \beta m - i+2) + \beta m - i+1, 1 + \beta m - i+1
\bigr\}
.
Fig. 3 now changes to Fig. 4.
Fig. 4. ith and (i+ 1)th TITSs.
Every associated lattice path enumerated by C1(\mu ) is uniquely generated in this manner. This
proves (2.1).
Step II. We now establish a 1 - 1 correspondence between the associated lattice paths enumerated
by C1(\mu ) and the n-color partitions enumerated by A1(\mu ). We do this by encoding each associated
lattice path as the sequence of weights of TITSs with each altitude of the TITS subscripted by the
length of the respective upper base. Thus, if we denote the two TITS in Fig. 4 by Pr and Qs,
respectively, then
P = (2i - 1) + 2(\beta m + \beta m - 1 + . . .+ \beta m - i+2) + \beta m - i+1,
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 1
54 M. GOYAL
r = \beta m - i+1 + 1,
Q = (2i+ 1) + 2(\beta m + \beta m - 1 + . . .+ \beta m - i+1) + \beta m - i,
s = \beta m - i + 1.
Clearly, the parity of P and r depends upon \beta m - i+1. If \beta m - i+1 is odd then both P and r are even
and when \beta m - i+1 is even then both P and r are odd. This proves that even parts are appearing with
even subscripts and odd with odd subscripts.
The weighted difference of these two parts is ((Qs - Pr)) = Q - P - r - s = 0.
The TITS with ordered pair \{ a, a\} corresponds to the part of the form aa or we can say kk in
the corresponding colored partition.
To see the reverse implication, we consider two n-color parts of a partition enumerated by A1(\mu ),
say, Pr and Qs with Q \geq P. Clearly r \leq P and s \leq Q.
Since Pr and Qs are the parts of n-color partition enumerated by A1(\mu ), weighted difference
equal ((Qs - Pr)) = 0 \Rightarrow Q - P - r - s = 0 \Rightarrow Q - s = P + r. Obviously, the part of the form kk
in the colored partition enumerated by A1(\mu ) will correspond to the TITS with ordered pair \{ k, k\}
or we can say \{ a, a\} .
Step III. Finally we establish a bijection between the weighted lattice paths enumerated by B1(\mu )
and the associated lattice paths enumerated by C1(\mu ). We do this by mapping each peak of weight a
and height b of a weighted lattice path enumerated by B1(\mu ) to a TITS with ordered pair \{ a, b\} of an
associated lattice path enumerated by C1(\mu ) and conversely. Under this mapping, all the conditions
on the weighted lattice paths enumerated by B1(\mu ) are translated to the conditions on the associated
lattice paths enumerated by C1(\mu ) and vice-versa. Hence this completes the bijection between the
weighted lattice paths enumerated by B1(\mu ) and the associated lattice paths enumerated by C1(\mu ).
2.3. Outline of the proofs of Theorems 2.2 – 2.5. Here, the changes required to prove the
remaining theorems are discussed briefly.
Theorem 2.2: This is treated in the same manner as Theorem 2.1. The only difference is now
the path begins with m TITSs with ith TITS having the ordered pair \{ 4i - 2, 2\} .
Theorem 2.3: In this case we observe that \beta 1, \beta 2, . . . , \beta m are 0 or 1 since the factor ( - q, q2)m
generates m nonnegative distinct multiples of (2i - 1), 1 \leq i \leq m.
Theorem 2.4: This is treated in the same manner as Theorem 2.3. The only difference is the alti-
tude of the first TITS is not increased since the factor ( - q, q2)m - 1 generates only m - 1 nonnegative
distinct multiples of (2i - 1), 1 \leq i \leq m - 1.
Theorem 2.5: An appeal to Theorem 2.2, the extra factor
q2m
1 - q2m+1
puts an SS of height 2 in
the beginning of the path or a TITS with ordered pair \{ a, a+ 2\} . Clearly, it will correspond to aa+2
or we can say ii+2 part of the corresponding colored partition.
3. Combinatorial interpretations by using antihook differences. Here, we apply antihook
differences to interpret mock theta functions (1.1) – (1.5) combinatorially. These results will further
extend Theorems 2.1 – 2.5 to 4-way combinatorial identities.
3.1. Main results.
Theorem 3.1. For \mu \geq 1, let D1(\mu ) denote the number of partitions of \mu such that (i) all
antihook differences on diagonal 0 are equal to 0 or 1, (ii) if [p, q] and [r, s] are any two consecutive
hooks such that p > r and q > s then q = r + 1 and (iii) if [p, q] is the last hook, then q = 0. Then
A1(\mu ) = B1(\mu ) = C1(\mu ) = D1(\mu ) for all \mu and
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ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 55
\infty \sum
\mu =1
D1(\mu )q
\mu =
\infty \sum
n=1
qn
2
(q; q2)n
= \Psi (q).
Theorem 3.2. For \mu \geq 0, let D2(\mu ) denote the number of partitions of \mu such that (i) all
antihook differences on diagonal 0 are equal to 0 or 1, (ii) there is no hook with rank \leq 0, (iii) if
[p, q] and [r, s] are any two consecutive hooks such that p > r and q > s, then q = r+1 and (iv) if
[p, q] is the last hook, then p \not = 0, q = 0. Then A2(\mu ) = B2(\mu ) = C2(\mu ) = D2(\mu ) for all \mu and
\infty \sum
\mu =0
D2(\mu )q
\mu =
\infty \sum
n=0
q2n
2
(q; q2)n
= F0(q).
Theorem 3.3. For \mu \geq 0, let D3(\mu ) denote the number of partitions of \mu such that (i) all
antihook differences on diagonal 0 are equal to 0 or 1, (ii) all hooks have rank 0 or 1, (iii) if [p, q]
and [r, s] are any two hooks such that p > r and q > s, then q = r + 1 and (iv) if [p, q] is the last
hook, then q = 0. Then A3(\mu ) = B3(\mu ) = C3(\mu ) = D3(\mu ) for all \mu and
\infty \sum
\mu =0
D3(\mu )q
\mu =
\infty \sum
n=0
qn
2
( - q; q2)n = \Phi 0(q).
Theorem 3.4. For \mu \geq 1, let D4(\mu ) denote the number of partitions of \mu such that (i) all
antihook differences on diagonal 0 are equal to 0 or 1, (ii) all hooks have rank 0 or 1, (iii) if [p, q]
and [r, s] are any two hooks such that p > r and q > s, then q = r + 1 and (iv) if [p, q] is the last
hook, then p = 0, q = 0. Then A4(\mu ) = B4(\mu ) = C4(\mu ) = D4(\mu ) for all \mu and
\infty \sum
\mu =1
D4(\mu )q
\mu =
\infty \sum
n=1
qn
2
( - q; q2)n - 1 = \Phi 1(q).
Theorem 3.5. For \mu \geq 0, let D5(\mu ) denote the number of partitions of \mu such that (i) all
antihook differences on diagonal 2 are equal to 1 or 2, (ii) there is no hook with rank \geq 2, (iii) if
[p, q] and [r, s] are any two consecutive hooks such that p > r and q > s, then q = r+1 and (iv) if
[p, q] is the last hook, then p = 0 or 2. Then A5(\mu ) = B5(\mu ) = C5(\mu ) = D5(\mu ) for all \mu and
\infty \sum
\mu =0
D5(\mu )q
\mu =
\infty \sum
n=0
q2n(n+1)
(q; q2)n+1
= F1(q).
Again the proofs of Theorems 3.1 – 3.5 are similar, we will provide a detailed proof of Theo-
rem 3.1 and an outline of the remaining proofs.
3.2. Proof of Theorem 3.1. Let \Pi be a partition enumerated by D1(\mu ). Let\Biggl(
p1 p2 . . . p\nu
q1 q2 . . . q\nu
\Biggr)
,
where p1 \geq p2 \geq . . . \geq p\nu \geq 0, q1 \geq q2 \geq . . . \geq q\nu \geq 0 and p1 + p2 + . . . + p\nu + q1 + q2 + . . .
. . . + q\nu + \nu = \mu , be the corresponding Frobenius symbol [15]. Then the antihook difference
conditions of Theorem 3.1 are equivalent to
pt \geq qt, (3.1)
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56 M. GOYAL
qt = pt+1 + 1, (3.2)
and
q\nu = 0. (3.3)
We now establish a bijection between the ordinary partitions enumerated by D1(\mu ) and the n-color
partitions enumerated by A1(\mu ). We do this by mapping each column
\biggl(
p
q
\biggr)
of the Frobenius symbol
to a single part gk of an n-color partition. The mapping is
\phi :
\Biggl(
p
q
\Biggr)
\rightarrow
\left\{ (p+ q + 1)q - p if p < q,
(p+ q + 1)p - q+1 if p \geq q.
(3.4)
The inverse mapping \phi - 1 is given by
\phi - 1 : gk \rightarrow
\left\{
\left( (g - k - 1)/2
(g + k - 1)/2
\right) if g \not \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2),
\left( (g + k - 2)/2
(g - k)/2
\right) if g \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2).
(3.5)
Now for any two adjacent columns
\biggl(
p r
q s
\biggr)
in the Frobenius symbol with \phi
\biggl(
p
q
\biggr)
= gk and
\phi
\biggl(
r
s
\biggr)
= hl as defined in (3.5), we have
((gk - hl)) =
\left\{
2q - 2r - 2 if p \geq q, r \geq s,
2p - 2r - 1 if p < q, r \geq s,
2q - 2s - 1 if p \geq q, r < s,
2p - 2s if p < q, r < s.
(3.6)
Clearly (3.1) and (3.4) imply the condition (i) of Theorem 1.1. Also (3.1), (3.3), and (3.4) ensures the
condition (ii) of Theorem 1.1 and then (3.2) and only the first line of (3.6) will imply the condition (iii)
of Theorem 1.1. To see the reverse implication we note that by condition (i) of Theorem 1.1 g \equiv k,
h \equiv l (\mathrm{m}\mathrm{o}\mathrm{d} 2) and so under \phi - 1
p - r =
1
2
((gk - hl)) + k, (3.7)
q - s =
1
2
((gk - hl)) + l, (3.8)
p - q = k - 1, (3.9)
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ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 57
q - r =
1
2
((gk - hl)) + 1. (3.10)
Now (3.7) and (3.8) by condition (iii) of Theorem 1.1 guarantee that pt > pt+1 and qt > qt+1.
(3.9) implies (3.1) and (3.10) by condition (iii) of Theorem 1.1 implies (3.2). Also, condition (ii)
of Theorem 1.1 and only the second line of (3.5) will imply (3.3). This completes the proof of
A1(\mu ) = D1(\mu ).
To illustrate the constructed bijections we give an example for \mu = 9 shown in the following
table.
Assosiated Frobenius
Partitions Lattice paths lattice paths Partitions symbols for
enumerated enumerated enumerated enumerated partitions
by A1(9) by B1(9) by C1(9)
by D1(9) enumerated
by D1(9)
86 + 11 7+2
\Biggl(
6 0
1 0
\Biggr)
73 + 22 5+3+1
\Biggl(
4 1
2 0
\Biggr)
51 + 31 + 11 3+3+3
\Biggl(
2 1 0
2 1 0
\Biggr)
3.3. Outline of the proofs of Theorems 3.2 – 3.5. Now let us discuss the essential steps to treat
the proofs of Theorems 3.2 – 3.5.
Theorem 3.2: In this case, the antihook difference conditions are equivalent to
pt \geq qt + 1,
qt = pt+1 + 1,
p\nu \not = 0,
and
q\nu = 0.
The map \phi is
\phi :
\Biggl(
p
q
\Biggr)
\rightarrow
\left\{ (p+ q + 1)q - p if p < q + 1,
(p+ q + 1)p - q+1 if p \geq q + 1
and \phi - 1 is given by
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58 M. GOYAL
\phi - 1 : gk \rightarrow
\left\{
\left( (g - k - 1)/2
(g + k - 1)/2
\right) if g \not \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2),
\left( (g + k - 2)/2
(g - k)/2
\right) if g \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2), k \not = 1.
Theorem 3.3: In this case, the antihook difference conditions are equivalent to
pt = qt or pt = qt + 1,
qt = pt+1 + 1,
and
q\nu = 0.
The map \phi is
\phi :
\Biggl(
p
q
\Biggr)
\rightarrow
\left\{ (p+ q + 1)q - p if p \not = q and p \not = q + 1,
(p+ q + 1)p - q+1 if p = q or p = q + 1
and \phi - 1 is given by
\phi - 1 : gk \rightarrow
\left\{
\left( (g - k - 1)/2
(g + k - 1)/2
\right) if g \not \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2), here k = 1 or 2,
\left( (g + k - 2)/2
(g - k)/2
\right) if g \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2), here k = 1 or 2.
Theorem 3.4: In this case, the antihook difference conditions are equivalent to
pt = qt or pt = qt + 1,
qt = pt+1 + 1,
and
p\nu = q\nu = 0.
The map \phi is
\phi :
\Biggl(
p
q
\Biggr)
\rightarrow
\left\{ (p+ q + 1)q - p if p \not = q and p \not = q + 1,
(p+ q + 1)p - q+1 if p = q or p = q + 1
and \phi - 1 is given by
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ON COMBINATORIAL EXTENSIONS OF SOME RAMANUJAN’S MOCK THETA FUNCTIONS 59
\phi - 1 : gk \rightarrow
\left\{
\left( (g - k - 1)/2
(g + k - 1)/2
\right) if g \not \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2), here k = 1 or 2,
\Biggl(
(g + k - 2)/2
(g - k)/2
\Biggr)
if g \equiv k (\mathrm{m}\mathrm{o}\mathrm{d} 2), here k = 1 or 2.
Theorem 3.5: Lastly, in this case, we observe that the antihook difference conditions are equiv-
alent to
pt \leq qt + 1,
pt = qt+1 + 3,
and
p\nu = 0 or 2.
The map \phi is
\phi
\Biggl(
p
q
\Biggr)
\rightarrow
\left\{ (p+ q + 1)q - p+3 if p \leq q + 1, p \not = 1,
(p+ q + 1)p - q - 1 if p > q + 1
and \phi - 1 is given by
\phi - 1 : gk \rightarrow
\left\{
\left( (g + k + 1)/2
(g - k - 3)/2
\right) if g \not \equiv k + 2 (\mathrm{m}\mathrm{o}\mathrm{d} 2),
\left( (g - k + 2)/2
(g + k - 4)/2
\right) if g \equiv k + 2 (\mathrm{m}\mathrm{o}\mathrm{d} 2), g \not = k.
4. Conclusion. In literature we find that a single basic series may have many combinatorial
interpretations in terms of different combinatorial objects (see, for instance, [1 – 3, 9, 11, 22 – 25]).
In this paper five mock theta functions of S. Ramanujan have been interpreted combinatorially using
associated lattice paths and antihook differences. After Ramanujan many more mock theta functions
of different orders have been found by several authors such as in [16, 18, 20, 21, 26, 27]. Gordon and
McIntosh [20] found some basic functions which they used to establish modular transformation for-
mulas for eighth order mock theta function. Now it would be of interest if these mock theta functions
could also be interpreted combinatorially using associated lattice paths and antihook differences.
References
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2. A. K. Agarwal, Antihook differences and some partition identities, Proc. Amer. Math. Soc., 110, 1137 – 1142 (1990).
3. A. K. Agarwal, Lattice paths and Rogers – Ramanujan type identities, Proc. 14th annual Conf. of the Ramanujan
Math. Soc., Banagalore, 31 – 39 (1999).
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№ 1 (2004).
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60 M. GOYAL
5. A. K. Agarwal, Lattice paths and mock theta functions, Proc. 6th Int. Conf. SSFA, Jaunpur, India, 95 – 102 (2005).
6. A. K. Agarwal, G. E. Andrews, Hook differences and lattice paths, J. Statist. Plann. Inference, 14, № 1, 5 – 14 (1986).
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Ser. A, 45, № 1, 40 – 49 (1987).
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209 – 228 (1989).
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10. A. K. Agarwal, M. Goyal, New partition theoretic interpretations of Rogers – Ramanujan identities, Int. J. Comb.
(2012).
11. A. K. Agarwal, M. Goyal, On 3-way combinatorial identities, Proc. Indian Acad. Sci. (Math. Sci.), to appear.
12. A. K. Agarwal, M. Rana, Two new combinatorial interpretations of a fifth order mock theta function, Indian Math.
Soc., Spec. Centenary Vol., 11 – 24 (1907 – 2007).
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(Math. Sci.), 122, № 1, 23 – 39 (2012).
15. G. E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc., 49, № 301 (1984).
16. G. E. Andrews, D. Hickerson, Ramanujan’s Lost notebook VII: The sixth order mock theta functions, Adv. Math., 89,
60 – 105 (1991).
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J., 66, № 8, 1131 – 1151 (2015).
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(2012).
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33 – 51 (2017).
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Received 21.07.16
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| id | umjimathkievua-article-2327 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:22:17Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/31/ff4fbca41b82565e4600208fd4de3431.pdf |
| spelling | umjimathkievua-article-23272020-01-27T11:49:51Z On combinatorial extensions of some Ramanujan’s mock theta functions О комбинаторные продолжения некоторых фиктивных тета-функций Рамануджана Про комбiнаторнi продовження деяких фiктивних тета-функцiй Рамануджана Goyal, M. Goyal, M. Goyal, M. фiктивні тета-функцiї Рамануджана кольорові розбиття граничні решітки Ramanujan’s mock theta functions mock theta functions $(n t)$--color partitions weighted lattice paths associated lattice paths anti-hook differences Five mock theta functions of S. Ramanujan are combinatorially interpreted by means of certain associated lattice path functions and antihook differences. These results provide new combinatorial interpretations of five mock theta functions of Ramanujan. Using a bijection between the associated lattice path functions and the $(n+t)$-color partitions and then between the associated lattice path functions and the weighted lattice path functions, we extend the works by Agarwal and Agarwal and Rana to five new 3-way combinatorial identities. These results are further extended to 4-way combinatorial identities by using bijection between the $(n+t)$-color partitions and the partitions with certain antihook differences. These interesting results present elegant combinatorial links between Ramanujan's mock theta functions, $(n+t)$-color partitions, weighted lattice paths, associated lattice paths, and antihook differences. &nbsp; Наведено комбінаторну інтерпретацію п'яти фіктивних тета-функцій С. Рамануджана за допомогою деяких асоційованих&nbsp; ратчастих функцій шляху та антигачкових різниць. Отримані результати дають нову комбінаторну інтерпретацію п'яти фіктивних тета-функцій Рамануджана. За допомогою взаємно однозначної відповідності між асоційованими ратчастими функціями шляху та $(n+t)$-кольоровими розбиттями, а також між асоційованими ратчастими функціями шляху та зваженими ратчастими функціями шляху узагальнено роботи Агарвала та Агарвала і Рана на випадок п'яти нових 3-шляхових комбінаторних тотожностей. Ці результати потім розширено на випадок 4-шляхових комбінаторних тотожностей за допомогою взаємно однозначної відповідності між $(n+t)$-кольоровими розбиттями та розбиттями з певними антигачковими різницями. Ці цікаві результати встановлюють елегантні комбінаторні зв'язки між фіктивними тета-функціями Рамануджана, $(n+t)$-кольоровими розбиттями, зваженими ратчастими шляхами, асоційованими ратчастими шляхами та антигачковими різницями. Institute of Mathematics, NAS of Ukraine 2020-01-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2327 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 1 (2020); 46-60 Український математичний журнал; Том 72 № 1 (2020); 46-60 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2327/1546 Copyright (c) 2020 M. Goyal |
| spellingShingle | Goyal, M. Goyal, M. Goyal, M. On combinatorial extensions of some Ramanujan’s mock theta functions |
| title | On combinatorial extensions of some Ramanujan’s mock theta functions |
| title_alt | О комбинаторные продолжения некоторых фиктивных тета-функций Рамануджана Про комбiнаторнi продовження деяких фiктивних тета-функцiй Рамануджана |
| title_full | On combinatorial extensions of some Ramanujan’s mock theta functions |
| title_fullStr | On combinatorial extensions of some Ramanujan’s mock theta functions |
| title_full_unstemmed | On combinatorial extensions of some Ramanujan’s mock theta functions |
| title_short | On combinatorial extensions of some Ramanujan’s mock theta functions |
| title_sort | on combinatorial extensions of some ramanujan’s mock theta functions |
| topic_facet | фiктивні тета-функцiї Рамануджана кольорові розбиття граничні решітки Ramanujan’s mock theta functions mock theta functions $(n t)$--color partitions weighted lattice paths associated lattice paths anti-hook differences |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2327 |
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