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Hurwitz type results for sums of squares and sums of triangular numbers
D. D. Somashekara,K. N. VIDYA 장전수학회 2019 Advanced Studies in Contemporary Mathematics Vol.29 No.4
Let rk(n) denote the number of representations of n as a sum of k squares and tk(n) denote the number of representations of n as a sum of k triangular numbers. Let r,(n) denote the number of representations of n as a sum of times a square and times another square and t,(n) denote the number of representations of n as a sum of times a triangular number and times another triangular number. We prove a number of results in which the generating function of r2(2k(an + b)), r4(2k(an + b)), r1,3(2k(an + b)), t2(2k(an + b)), t1,2(2k(an + b)), for various integer values of k, a and b, is a simple infinite product. We also obtain a relation between t1,3(3k+1n + 3k + 3k−1 · · · +3 + 1) and t1,3(n).
On some identities of Ramanujan found in his lost notebook
D.D. Somashekara,D. Mamta 장전수학회 2008 Advanced Studies in Contemporary Mathematics Vol.16 No.2
In this paper we obtain an interesting identity using the well-knownq-binomial the-orem and Heine's transformation. Using this identity, we derive some identities ofRamanujan found in his lost notebook and also obtain some identities which areanalogous to the identities of Ramanujan along with some special cases.
HURWITZ TYPE RESULTS FOR CERTAIN REPRESENTATIONS OF INTEGERS AS SUMS OF SQUARES
D. D. Somashekara,THULASI.M.B. 장전수학회 2022 Proceedings of the Jangjeon mathematical society Vol.25 No.3
Let rα,β(n) denote the number of representations of n as a sum of α times a square and β times another square. In the recent past, a number of authors have obtained Hurwitz type results for representations of integers as sums of squares. Motivated by their works, in this paper we prove many results in which the generating function of r2,3(λk(an + b)), r2,4(an + b) and r1,5(λk(an + b)) for various non-negative integer values of λ, k, a and b are infinite products. To obtain our main results we use the theta function identities of Ramanujan found in chapter 16 of his second notebook.
On some applications of the new forms of the reciprocity theorems
D. D. Somashekara,S. L. Shalini 장전수학회 2014 Advanced Studies in Contemporary Mathematics Vol.24 No.3
In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we present a number of interesting applications of the new symmetric forms of the reciprocity theorems.
On square sum labelings of graphs
D. D. Somashekara,C. R. Veena 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.1
Abstract. Let G = (V,E) be an (n,m)-graph. A graph G is said to admit a square sum labeling, if its vertices can be labeled with distinct non-negative integers so that the values on the edges, obtained as the sum of the labels of their end vertices, are the first m squares. In this paper we show that certain classes of graphs admit square sum labeling and also that certain classes of graphs do not admit square sum labeling.
On pentagonal sum labelings of graphs
D. D. Somashekara,C. R. Veena 장전수학회 2012 Advanced Studies in Contemporary Mathematics Vol.22 No.3
Let G = (V,E) be an (n,m)-graph. A graph G is said to admit a pentagonal sum labeling, if its vertices can be labeled with distinct non-negative integers so that the values on the edges, obtained as the sum of the labels of their end vertices, are the first m pentagonal numbers. In this paper we show that certain classes of graphs admit pentagonal sum labeling and also that certain classes of graphs do not admit pentagonal sum labeling.
On the reciprocity theorem of Ramanujan and its Generalizations
D. D. Somashekara,K. N. Murthy,S. L. Shalini 장전수학회 2012 Proceedings of the Jangjeon mathematical society Vol.15 No.3
In his lost notebook, Ramanujan has stated a beautiful two variable reciprocity theorem. In the recent past, three and four variable generalizations of the reciprocity theorem of Ramanujan were established. In this paper, we give an unified approach to the proofs of all the three reciprocity theorems. Some new q-gamma, q-beta and eta-function identities were also be deduced from the four variable reciprocity theorem.
ON SOME PARTIAL THETA FUNCTION IDENTITIES OF RAMANUJAN FOUND IN HIS LOST NOTEBOOK
D. D. Somashekara,T. Kim,H. I. Kwon,S. L. Shalini 장전수학회 2015 Advanced Studies in Contemporary Mathematics Vol.25 No.1
Ramanujan has recorded several identities involving partial theta functions in his lost notebook. In 1981, Andrews has given proofs for sixteen of those identities by employing his general identity. In this paper, we obtain ten of those identities by an alternate method. We also deduce several interesting special cases.
On skew-generalised energy of digraphs
D. D. Somashekara,H. E. Ravi 장전수학회 2020 Proceedings of the Jangjeon mathematical society Vol.23 No.4
In this paper we introduce the concept of skew-generalised energy of directed graphs. We then obtain upper and lower bounds for skew-generalised energy of digraphs. Then we compute the skew-generalised energy of some graphs such as star digraph, complete bipartite digraph, the (Sm ^ P2) digraph, (n, 2n − 3) strong vertex graceful digraph and a crown digraph.
ON THE SKEW ENERGY OF SOME UNITARY CAYLEY DIGRAPHS
C. Adiga,H.N. Ramaswamy,D.D. Somashekara,Z. Khoshbakht 장전수학회 2010 Advanced Studies in Contemporary Mathematics Vol.20 No.2
In this paper we introduce Cayley digraph Dn, n > 1. We show that when n = 3(mod 4) the skew eigenvalues of Dn are the Gauss sums associated with the quadratic character and we also compute its skew energy.