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Agarwal, A. K.
- An Extension of Euler’s Theorem
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Authors
Affiliations
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
Source
The Journal of the Indian Mathematical Society, Vol 70, No 1-4 (2003), Pagination: 17-24Abstract
Euler’ s classical partition identity “The number of partitions of an integer v into distinct parts is equal to the number of its partitions into odd parts” is extended to eight more combinatorial functions. This results in a 10-way combinatorial identity which implies 45 combinatorial identities in the usual sense. Euler’s identity is just one of them.
- Alternate Terms in Lucas Sequence
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Authors
A. K. Agarwal
1,
B. Dutta
2
Affiliations
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
2 Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Khanapara, Guwahati-781022, IN
1 Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-160014, IN
2 Mathematical Sciences Division, Institute of Advanced Study in Science and Technology, Khanapara, Guwahati-781022, IN
Source
The Journal of the Indian Mathematical Society, Vol 66, No 1-4 (1999), Pagination: 27-32Abstract
In this paper we study the properties of L2n+1 which is the Lucas number of order 2n+1. Several properties like generating functions, recurrence relations, summation formulas and (q-analogues of L2n+1 were found by Agarwal in [1,2,3]. Here we obtain hypergeometric form, Integral representation and several congruence properties and identities for these numbers. Congruence properties are used to establish a theorem on periodicity of the sequence {L2n+1}n∞=0.- The Bailey Lattice
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1+∑qn2/(1-q)(1-q2)...(1-qn) (1.1)
=∑1/(1-q5n+1)(1-q5n+2)2
1+∑qn2+n/(1-q)(1-q2)...(1-qn) (1.2)
=∑1/(1-q5n+2)(1-q5n+3)
These are equivalent respectively to the following combinatorial identities:
The number of partitions of n into parts with difference at least 2 equals the number of partitions of n into parts congruent to ±1, modulo 5.
Authors
Affiliations
1 Pennsylvania State University, Mont Alto, US
2 Pennsylvania State University, University Park, US
1 Pennsylvania State University, Mont Alto, US
2 Pennsylvania State University, University Park, US
Source
The Journal of the Indian Mathematical Society, Vol 51, No 1-2 (1987), Pagination: 57-73Abstract
The Rogers-Ramanujan identities [5; ch. 7] are given analytically by the following formulae: (|q|<1)1+∑qn2/(1-q)(1-q2)...(1-qn) (1.1)
=∑1/(1-q5n+1)(1-q5n+2)2
1+∑qn2+n/(1-q)(1-q2)...(1-qn) (1.2)
=∑1/(1-q5n+2)(1-q5n+3)
These are equivalent respectively to the following combinatorial identities:
The number of partitions of n into parts with difference at least 2 equals the number of partitions of n into parts congruent to ±1, modulo 5.