Nayandeep Deka Baruah

Tezpur University · Department of Mathematical Sciences

We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfra...

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prim...

A partition is said to be [Formula: see text]-regular if none of its parts is a multiple of [Formula: see text]. Let [Formula: see text] denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of [Formula: see text]. This function has also close connections to representation theory and combinatorics. In this pap...

Recently, Gireesh, Ray, and Shivashankar studied an analog, $\overline{a}_t(n)$, of the $t$-core partition function, $c_t(n)$. In this paper, we study the function $\overline{a}_5(n)$ in conjunction with $c_5(n)$ as well as another analogous function $\overline{b}_5(n)$. We also find several arithmetic identities for $\overline{a}_5(n)$ and $\overl...

Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical M\"{o}bius function as well as Euler's totient. In this paper, we define a counting function $T_k^r(m)$ on the set of $n$-color partitions of $m$ for given positive integers $k, r$ and relate the function with the $n$-color partition fu...

Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a generalized result that gives such relations for $s$-gonal numbers for any integer $s\geq3$.

Recently, Gireesh, Ray, and Shivashankar studied an analog, a t (n), of the t-core partition function, c t (n). In this paper, we study the function a 5 (n) (A053723) in conjunction with c 5 (n) (A368490), as well as another analogous function b 5 (n) (A368495). We also find several arithmetic identities for a 5 (n) and b 5 (n).

We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$ , $p(n)=0$ for $n<0$ ), then for all nonnegative integers m , $$ \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))...

Partitions wherein the even parts appear in two different colors are known as cubic partitions. Recently, Merca introduced and studied the function A(n), which is defined as the difference between the number of cubic partitions of n into an even number of parts and the number of cubic partitions of n into an odd number of parts. In particular, usin...

In this article, we refine a result of Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of partitions of n into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three further congru...

Recently, Jha has found identities that connect certain sums over the divisors of n to the number of representations of n as a sum of squares and triangular numbers. In this note, we state a generalized result that gives such relations for s-gonal numbers for any integer s ≥ 3.

In the eleventh paper in the series on MacMahon’s partition analysis, Andrews and Paule introduced the [Formula: see text]-elongated partition diamonds. Recently, they revisited the topic. Let [Formula: see text] count the partitions obtained by adding the links of the [Formula: see text]-elongated plane partition diamonds of length [Formula: see t...

We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then $$R(q)R(q^4)&=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})}\\\intertext{and}\dfrac{1}{R(q^{2})R(q^{3})}&+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfr...

Partitions wherein the even parts appear in two different colours are known as cubic partitions. Recently, Merca introduced and studied the function $A(n)$, which is defined as the difference between the number of cubic partitions of $n$ into an even number of parts and the number of cubic partitions of $n$ into an odd number of parts. In particula...

We find an interesting refinement of a result due to Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number n equals the number of partitions of n into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three...

Recently, da Silva and Sellers [11] studied the arithmetical properties of the coefficients pξ(n) of the third order mock theta function ξ(q), which was introduced by Gordon and McIntosh and is defined by ξ(q)=1+2∑n=1∞q6n2-6n+1q;q6nq5;q6n=:∑n=0∞pξ(n)qn.Da Silva and Sellers discovered several congruences modulo 3, 4, 5, 8, and 9 for pξ(n). In this p...

For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.

In the eleventh paper in the series on MacMahons partition analysis, Andrews and Paule [1] introduced the $k$ elongated partition diamonds. Recently, they [2] revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the $k$ elongated plane partition diamonds of length $n$. Andrews and Paule [2] obtained several generat...

Let [Formula: see text] and [Formula: see text] count the 8-colored partitions of [Formula: see text] into odd parts and the [Formula: see text]-color partitions of [Formula: see text] into odd parts, respectively, where the multiples of [Formula: see text] can appear in [Formula: see text] colors and the others can appear in [Formula: see text] co...

Recently, Gireesh, Shivashankar, and Naika [11] found some infinite classes of congruences for the 3- and the 9-regular cubic partitions modulo powers of 3. We extend their study to all the $3^k$-regular cubic partitions. We also find new families of congruences.

We show that the series expansions of certain q-products have matching coefficients with their reciprocals. Several of the results are associated to Ra-manujan's continued fractions. For example, let R(q) denote the Rogers-Ramanujan continued fraction having the well-known q-product repesentation R(q) = (q; q 5) ∞ (q 4 ; q 5) ∞ (q 2 ; q 5) ∞ (q 3 ;...

Let $b_{\ell;3}(n)$ denote the number of $\ell$-regular partitions of $n$ in 3 colours. In this paper, we find some general generating functions and new infinite families of congruences modulo arbitrary powers of $3$ when $\ell\in\{9,27\}$. For instance, for positive integers $n$ and $k$, we have\begin{align*}b_{9;3}\left(3^k\cdot n+3^k-1\right)&\e...

Recently, Chan and Wang studied the fractional partition function and found several infinite classes of congruences satisfied by the corresponding coefficients. In this paper, we find new families of congruences modulo powers of primes using the Rogers-Ramanujan continued fraction and some dissection formulae of certain q-products. We also find ana...

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_...

Recently, Merca and Schmidt found some decompositions for the partition function p(n) in terms of the classical Möbius function as well as Euler’s totient. In this paper, we define a counting function T_r ^k(m) on the set of n-color partitions of m for given positive integers k, r and relate the function with the n-color partition function and othe...

Let N(c1,c2,…,ck;n),r(c1,c2,…,ck;n), and t(c1,c2,…,ck;n) count the representations of a positive integer n in the quadratic forms c1x12+x1x2+x22+c2x32+x3x4+x42+⋯+ck(x2k-12+x2k-1x2k+x2k2),c1x12+c2x22+⋯+ckxk2,c1x1(x1+1)/2+c2x2(x2+1)/2+⋯+ckxk(xk+1)/2,respectively, where ci’s and xi’s are integers. Using Ramanujan’s theta functions φ(q), ψ(q), Borweins...

Recently, Hirschhorn proved that, if ∑n=0∞anqn:=(-q,-q4;q5)∞(q,q9;q10)∞3and ∑n=0∞bnqn:=(-q2,-q3;q5)∞(q3,q7;q10)∞3,then a5n+2=a5n+4=0 and b5n+1=b5n+4=0. Motivated by the work of Hirschhorn, Tang proved some comparable results including the following: If ∑n=0∞cnqn:=(-q,-q4;q5)∞3(q3,q7;q10)∞and ∑n=0∞dnqn:=(-q2,-q3;q5)∞3(q,q9;q10)∞,then c5n+3=c5n+4=0an...

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences mo...

Recently, Lin introduced two new partition functions \(\hbox {PD}_{\mathrm{t}}(n)\) and \(\hbox {PDO}_{\mathrm{t}}(n)\), which count the total number of tagged parts over all partitions of n with designated summands and the total number of tagged parts over all partitions of n with designated summands in which all parts are odd. Lin also proved som...

Relations involving the Rogers-Ramanujan continued fractions $R(q),$ $R(q^3 ),$ and $R(q^4)$ are used to find new generating functions and congruences modulo 5 and 25 for 3-core, 4-core, 4-regular, and colored partition functions.

Let N(a, b, c, d; n) be the number of representations of n as ax2+by2+cz2+dw2 and T(a, b, c, d, n) be the number of representations of n as \(a\frac{{X(X + 1)}}{2} + b\frac{{Y(Y + 1)}}{2} + c\frac{{Z(Z + 1)}}{2} + d\frac{{W(W + 1)}}{2}\) , where a, b, c, d are positive integers, n, X, Y, Z, W are nonnegative integers, and x, y, z, w are integers. R...

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(10 n+ 9) in terms of q-products. As corollaries, we deduce the congruences a(50n+1...

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third order mock theta functions $\omega(q)$ and $\nu(q)$. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest parts functions and deduce several new congruences modulo powers of...

Recently, M. D. Hirschhorn proved that, if $\sum_{n=0}^\infty a_nq^n := (-q,-q^4;q^5)_\infty(q,q^9;q^{10})_\infty^3$ and $\sum_{n=0}^\infty b_nq^n:=(-q^2,-q^3;q^5)_\infty(q^3,q^7;q^{10})_\infty^3$, then $a_{5n+2}=a_{5n+4}=0$ and $b_{5n+1}=b_{5n+4}=0$. Motivated by the work of Hirschhorn, D. Tang proved some comparable results including the followin...

Recently, Andrews, Dixit and Yee introduced partition functions associated with Ramanujan/Watson third-order mock theta functions [Formula: see text] and [Formula: see text]. In this paper, we find several new exact generating functions for those partition functions as well as the associated smallest part functions and deduce several new congruence...

On page 3 of his lost notebook, Ramanujan defines the Appell-Lerch sum $$\phi(q):=\sum_{n=0}^\infty \dfrac{(-q;q)_{2n}q^{n+1}}{(q;q^2)_{n+1}^2},$$ which is connected to some of his sixth order mock theta functions. Let $\sum_{n=1}^\infty a(n)q^n:=\phi(q)$. In this paper, we find a representation of the generating function of $a(10n+9)$ in terms of...

Recently, Lin introduced two new partition functions $\textup{PD}_\textup{t}(n)$ and $\textup{PDO}_\textup{t}(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some...

Let (Formula presented.) denote the number of partitions of a non-negative integer into distinct (or, odd) parts. We find exact generating functions for (Formula presented.), (Formula presented.) and (Formula presented.). We deduce some congruences modulo 5 and 25. We employ Ramanujan’s theta function identities and some identities for the Rogers–R...

Paper 19: Nayandeep Deka Baruah, Bruce C. Berndt and Heng Huat Chan, “Ramanujan’s series for 1/π: A survey,” American Mathematical Monthly, vol. 116 (2009), p. 567–587. Copyright 2009 Mathematical Association of America. All Rights Reserved. Synopsis: In this piece, the authors discuss some formulas for 1/π originally discovered by Ramanujan. One o...

We find several new congruences for \(\ell \)-regular partitions for \(\ell \in \{5,6,7,49\}\) and also find alternative proofs of the congruences for 10- and 20-regular partitions which were proved earlier by Carlson and Webb (Ramanujan J 33:329–337, 2014) by using the theory of modular forms. We use certain p-dissections of \((q;q)_{\infty }\), \...

Let p k (n) be the number of 2-color partitions of n where one of the colors appears only in parts that are multiples of k. In this paper, we find some interesting congruences modulo 5 for p k (n) for k ∈ {2, 3, 4} by employing Ramanujan's theta function identities and some identities for the Rogers– Ramanujan continued fraction. The congruence for...

We present the generating function for $$c\phi _6(n)$$cϕ6(n), the number of generalized Frobenius partitions of $$n$$n with $$6$$6 colors, in terms of Ramanujan’s theta functions and exhibit $$2$$2, and $$3$$3-dissections of it that yield the congruences $$c\phi _6(2n+1)\equiv 0~(\text {mod}~4)$$cϕ6(2n+1)≡0(mod4), $$c\phi _6(3n+1)\equiv 0~(\text {m...

We find new partition identities arising from Ramanujan’s formulas of multipliers. Several of the identities are for overpartitions, overpartition pairs, and (Formula presented.)-regular partitions.

We find several new congruences modulo 2 for 11 dots bracelet partitions and congruences modulo and for k dots bracelet partitions for any prime and with and , respectively. We also find a relation between 5-core partitions and Ramanujan's famous tau function and several new congruences modulo 9 and 25 for 3- and 5-core partitions, respectively.

We find several new parity results for 7- and 23-regular partitions.

Let denote the number of bipartitions of n that are 3-cores. By employing Ramanujan's simple theta function identities, we prove that , where denotes the sum of the positive divisors of n. We also find several infinite families of arithmetic identities and congruences for , which include generalizations of some recent results on by B.L.S. Lin (2014...

In this paper we derive some new identities involving Ramanujan’s Cubic continued fraction G(q) and we give a new proof of a modular identity relating G(q) and G(q3 ).

Recently, Andrews defined the combinatorial objects which he called singular overpartitions and proved that these singular overpartitions, which depend on two parameters k and i, can be numerated by the function Ck,i(n) which gives the number of overpartitions of n in which no part is divisible by k and only parts ≡ ±i (mod k) may be overlined. He...

We find several new parity results for broken 5-diamond, 7-diamond and 11-diamond partitions.

Recently, Sandon and Zanello (Ramanujan J 33: 83–120, 2014) conjectured 29 highly non-trivial colored partition identities. In this paper, we establish 17 of them and prove analogous colored partition identities of the remaining 12 conjectural identities by using the theory of Ramanujan’s theta functions. We also present some new colored partition...

We find infinite families of arithmetic identities for self-conjugate 5-cores and 7-cores by employing Ramanujan’s simple theta function identities.

We prove that if u(n) denotes the number of representations of a nonnegative integer n in the form x2 + 3y2 with x, y ∈ ℤ, and a3(n) is the number of 3-cores of n, then u(12n+ 4) = 6a3(n). With the help of a classical result by L. Lorenz in 1871, we also deduce that a3(n) = d1,3(3n + 1) - d2,3(3n + 1), where dr,3(n) is the number of divisors of n c...

Relations between doubled distinct and self-conjugate t-cores for some small values of t are derived. By employing Ramanujan's simple theta function identities, several infinite families of arithmetic identities for doubled distinct t-cores for t = 3, 4, …, 10 are found.

We present some identities and congruences for the general partition function p r (n). In particular, we deduce some known identities for Ramanujan’s tau function and find simple proofs of Ramanujan’s famous partition congruences for modulo 5 and 7. Our emphasis throughout this paper is to exhibit the use of Ramanujan’s theta functions to generate...

Let u(n) and v(n) be the number of representations of a nonnegative integer n in the forms x 2+4y 2+4z 2 and x 2+2y 2+2z 2, respectively, with x,y,z∈ℤ, and let a 4(n) and r 3(n) be the number of 4-cores of n and the number of representations of n as a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we...

In this paper, we study the partition function p[cldm](n)p_{[c^{l}d^{m}]}(n) defined by ån=0¥p[cldm](n)qn=(qc;qc)¥-l(qd;qd)¥-m\sum_{n=0}^{\infty}p_{[c^{l}d^{m}]}(n)q^{n}=(q^{c};q^{c})_{\infty}^{-l}(q^{d};q^{d})_{\infty}^{-m} and prove some analogues of Ramanujan’s partition identities. We also deduce some interesting partition congruences. Keyword...

In this paper, the authors present twenty-five analogues of Jacobi’s two-square theorem which involve squares, triangular numbers, pentagonal numbers, heptagonal numbers, octagonal numbers, decagonal numbers, hendecagonal numbers, dodecagonal numbers, and octadecagonal numbers.

Ramanujan recorded two beautiful theta function identities on p. 310 of his second notebook. Employing Ramanujan’s identities, we deduce several results on the number of representations of a number as a sum of three squares and as a sum of three triangular numbers previously found by M. D. Hirschhorn and J. A. Sellers [Acta Arith. 77, No. 3, 289–30...

We present some congruences involving the functions cϕ4(n) and cϕ¯4(n) which denote, respectively, the number of generalized Frobenius partitions of n with 4 colors and 4-order generalized Frobenius partitions of n with 4 colors.

By using certain representations for Eisenstein series, we find new hypergeometric-like series for 1/π 2 arising from Ramanujan's theory of elliptic functions to alternative base 3.

Using certain representations for Eisenstein series, we uniformly derive several Ramanujan-type series for 1/π. KeywordsEisenstein series-Theta-functions-Modular equations Mathematics Subject Classification (2000)33C05-33E05-11F11-11R29

We present alternative proofs of some of Ramanujan’s theta function identities associated with the modular equations of composite degree 15. Along the way we also find some new theta-function identities. We also give simple proofs of his modular equations of degree 15. KeywordsTheta function-elliptic integral-modular equation-multiplier

Using hypergeometric identities and certain representations for Eisenstein series, we uniformly derive several new series representations for 1/π2.

When we pause to reflect on Ramanujan's life, we see that there were certain events that seemingly were necessary in order that Ramanujan and his mathemat- ics be brought to posterity. One of these was V. Ramaswamy Aiyer's founding of the Indian Mathematical Society on 4 April 1907, for had he not launched the Indian Mathematical Society, then the...

In this paper, we establish several theorems for the explicit evaluations of Ramanujan-Göllnitz-Gordon continued fraction by using some parameterizations of Ramanujan’s theta-functions.

Let rk (n) and tk (n) denote the number of representations of an integer n as a sum of k squares, and as a sum of k triangular numbers, respectively. We prove that t8(n) = 1/210 × 32 (r8+8) - 16r8 (2n + 2)) and therefore the study of the sequence t8(n) is reduced to the study of subsequences of r8(n). We give an additional 21 analogous results for...

Using certain representations for Eisenstein series, we derive several of Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions.

In this paper, we find new proofs of modular relations for the Göllnitz-Gordon functions established earlier by S.-S. Huang and S.-L. Chen. We use Schröter’s formulas and some simple theta-function identities of Ramanujan to establish the relations. We also find some new modular relations of the same nature.

We define the nonic Rogers–Ramanujan-type functions D(q), E(q) and F(q) and establish several modular relations involving these functions, which are analogous to Ramanujan's well known forty identities for the Rogers–Ramanujan functions. We also extract partition theoretic results from some of these relations.

The authors show that certain theta function identities of Schröter and Ramanujan imply elegant partition identities.

We define two parameters $g_{k,n}$ and $g^\prime_{k,n}$ involving Ramanujan's theta-functions $\psi(q)$ for any positive real numbers $k$ and $n$. We study several properties of these parameters and find some explicit values of $\psi(q)$ and quotients of $\psi(q)$ and of $\phi(q)$. This work is a sequel to some recent works by J. Yi.

We show that certain modular equations and theta function identities of Ramanujan imply elegant parti- tion identities. Several of the identities are for t-cores.

Beginning with S. Ramanujan’s epic paper, “Modular equations and approximations to π” [Quart. J. Math. 45, 350–372 (1914; JFM 45.1249.01)], we describe Ramanujan’s series for 1/π and later attempts to prove them. Generalizations, analogues, and consequences of Ramanujan’s series are discussed.

By employing a method of parameterizations for Ramanujan's theta-functions, we find several modular relations and explicit values of the Ramanujan-Selberg continued fractions.

We provide alternative derivations of theta function identities associated with modular equations of degree 5. We then use the identities to derive the corresponding modular equations.

Bruce C. Berndt et al. [with H. H. Chan, Mathematika 42, No. 2, 278–294 (1995; Zbl 0862.33016), with Chan and L.-C. Zhang, Acta Arith. 73, No. 1, 67–85 (1995; Zbl 0843.11007)] and Soon-Yi Kang [Acta Arith. 90, No. 1, 49–68 (1999; Zbl 0933.11003), Ramanujan J. 3, No. 1, 91–111 (1999; Zbl 0930.11025)] have proved many of Ramanujan’s formulas for the...

We apply new and old eta-function identities involving eight arguments to find some new explicit values of Ramanujan’s continued fractions and two parameters $\lambda_n$ and $\mu_n$ connected with Ramanujan’s cubic theory of elliptic functions.

In 2001, Jinhee Yi found many explicit values of the famous Rogers–Ramanujan continued fraction by using modular equations and transformation formulas for theta-functions. In this paper, we use her method to ÿnd some general theorems for the explicit evaluations of Ramanujan's cubic continued fraction.

We give elementary proofs of seven Schläfli-type “mixed” modular equations recorded by Ramanujan on p. 86 of his first notebook. Previously, these equations were proved by Berndt by using the theory of modular forms. In the process, we also found three new Schläfli-type mixed modular equations of the same nature.

We give elementary proofs of seven Schlafli-type "mixed" modular equations recorded by Ramanujan on p. 86 of his first notebook. Previously, these equations were proved by Berndt by using the theory of modular forms. In the process, we also found three new Schlafli-type mixed modular equations of the same nature.

In this paper we present two new identities providing relations between Ramanujan's cubic continued fraction G(q) and the two continued fractions G(q5) and G(q7).

The purpose of this paper is to establish Ramanujan’s class invariant G 217 via modular equations of degrees 7 and 31. We also show how Ramanujan’s Schläfli-type modular equations of composite degrees can be combined with those of prime degrees to obtain some of his class invariants.

We establish several modular relations involving two functions analogous to the Rogers-Ramanujan functions. These relations are analogous to Ramanujan’s famous forty identities for the Rogers-Ramanujan functions. Also, by the partitions, we extract partition theoretic interpretations from some of our relations.