Introduction

The great Indian Mathematician Srinivasa Ramanujan introduced the concept of Ramanujan Summation as one of the methods of sum ability theory where he gave a nice formula for summing powers of positive integers which is connected to Bernoulli numbers and Riemann zeta function. In this paper, I will present the formulas given by Ramanujan and use them to derive the Ramanujan summation formulas for powers of triangular as well as Pronic numbers. In this aspect, I had derived new formulas for determining such summation values.

Definitions and Formulas

The first few values of Bernoulli Numbers are given by



From the above values, we observe that except for B₁, B_n = 0 for all odd values of n.

Let  be a divergent series of real numbers. The Ramanujan summation abbreviated as RS (see [1]) of  is defined by



Srinivasa Ramanujan proved a formula connecting Riemann zeta function with Bernoulli numbers (for proof see [2]). The formulas called as Ramanujan Summation were given by



 

Here r is any positive integer and  is the Riemann zeta function. Using these definitions and formulas, I will prove some new results in this paper.

Powers of Triangular Numbers

In this section, I will determine a formula expressing positive integral powers of triangular numbers.

Theorem 1

If r is any natural number, then the rth power of kth triangular number is given by



Proof: By (2.1) and using binomial expansion for positive integers we have



This proves (3.1) and completes the proof.

Theorem 2

The Ramanujan summation for positive integral powers of triangular numbers is given by

Proof: Using (2.1), (2.5), (2.6), (2.7) and (3.1), we have



 

This completes the proof.

Powers of Pronic Numbers

In this section, I will determine a formula expressing positive integral powers of Pronic numbers.

Theorem 3

If r is any natural number, then the rth power of kth triangular number is given by



Proof: By (2.3) and using binomial expansion for positive integers we have



This proves (4.1) and completes the proof.

Theorem 4

The Ramanujan summation for positive integral powers of Pronic numbers is given by



Proof: First, we notice by definition that the Pronic numbers are exactly twice the corresponding triangular numbers. That is,



Hence, by (3.2) of theorem 2, we have



 

This completes the proof.

Computation of Ramanujan Summation Values

In this section, I will use results obtained in theorems 2 and 4 to evaluate Ramanujan summation of certain divergent series.

Corollary 1



Proof: Taking r = 1, 2, 3, 4 in (3.2), we get



This completes the proof.

Corollary 2



Proof: Taking r = 1, 2, 3, 4 in (4.2), we get



This completes the proof.

Conclusion

Srinivasa Ramanujan presented Ramanujan summation formulas for positive integral powers of natural numbers in terms of Bernoulli numbers. In this paper, I had extended this idea to determine Ramanujan summation for positive integral powers of triangular and Pronic numbers. Pronic numbers are also called as Rectangular numbers or Oblong numbers. For doing this, I had made use of the formulas provided by Ramanujan as given in (2.6) and (2.7). Using these two formulas and Bernoulli numbers, I had proved two new formulas for determining Ramanujan summation values for positive integral powers of triangular and Pronic numbers through (3.2) and (4.2) respectively. Two corollaries were presented in section 5 to compute the Ramanujan summation values of first four powers of triangular and Pronic numbers through equations (5.1) to (5.8). The results proved through theorems 2 and 4 are new and would provide further scope for understanding the meaning and connection between sum ability theory and Riemann zeta functions with respect to analytic continuation.

Acknowledgements

The author acknowledges the encouragement and help rendered by African Moon University, South West Africa and the USA for supporting to publication of this research paper as part of D.Math. Degree.

Funding Source

There is no funding or financial support provided for doing this research work.

References

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