Generalization of Euler and Ramanujan’s Partition Function

Mohajan, Haradhan (2015): Generalization of Euler and Ramanujan’s Partition Function. Published in: Asian Journal of A pplied Science and Engineering , Vol. 4, No. 3 (12 November 2015): pp. 167-190.

Abstract

The theory of partitions has interested some of the best minds since the 18th century. In 1742, Leonhard Euler established the generating function of P(n). Godfrey Harold Hardy said that Srinivasa Ramanujan was the first, and up to now the only, mathematician to discover any such properties of P(n). In 1981, S. Barnard and J.M. Child stated that the different types of partitions of n in symbolic form. In this paper, different types of partitions of n are also explained with symbolic form. In 1952, E. Grosswald quoted that the linear Diophantine equation has distinct solutions; the set of solution is the number of partitions of n. This paper proves theorem 1 with the help of certain restrictions. In 1965, Godfrey Harold Hardy and E. M. Wright stated that the ‘Convergence Theorem’ converges inside the unit circle. Theorem 2 has been proved here with easier mathematical calculations. In 1853, British mathematician Norman Macleod Ferrers explained a partition graphically by an array of dots or nodes. In this paper, graphic representation of partitions, conjugate partitions and self-conjugate partitions are described with the help of examples.

• Item Type: MPRA Paper
• Language: English
• Keywords: Ferrers and Young diagram, generating function, partitions, Ramanujan
• Subjects: C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C60 - General

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