Additive and Multiplicative Operations on Set of Polygonal Numbers

In this paper, the focus on generating k-gonal numbers P (k, n) \(=\) \(\{ {array}{r} n/2 ( k - 3 )( n - 1 )+( n + 1 ) \ for\ k > 2,n ≥ 0 \\ n/2 ( k - 3 )( n + 1 )+( n - 1 ) \ for\ k > 2,n < 0 {array} \}\). Also introduce to define additive (+) and multiplicative (*) operations on Sets of k-gonal numbers. In particular, concerning addition (+), \(p( k,m )+ p( k,n )= p( k,m + n )-( k - 2 )mn\ \ forsome\ integer\ k > 2\). Also, concerning multiplication (*), \(p( k,m )*p( k,n )= p( k,mn )+( k - 4 )( k - 2 )mn/4( m - 1 )( n - 1 )\ forsome\ integer\ \ k > 2\).Also, summation of any two different k-gonal is \(p( k₁,n )+ p( k₂,n )= n( n + 1 )+n( n - 1 )/2 ( k₁+k₂ )- 6 \ for\ k₁,k₂> 2\). Also, I applied above properties on some Sets of Polygonal numbers, which are generated by replacing integer k with 3,4,5,6,7 and 8. Also, introduced to study of Repeated steps of Residues of the above Sets of Polygonal numbers generated by k with 3,4,5,6,7 and 8.