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General convolution sums involving Fibonacci m-step numbers

Robert Frontczak, Karol Gryszka

Abstract

In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.

General convolution sums involving Fibonacci m-step numbers

Abstract

In this paper, using a generating function approach, we derive several new convolution sum identities involving Fibonacci m-step numbers. As special instances of the results derived herein, we will get many new and known results involving Fibonacci, Tribonacci, Tetranacci and Pentanacci numbers. In addition, we establish some general results providing insights into the inner structure of such convolutions. Finally, some mixed convolutions involving Fibonacci m-step numbers and Jacobsthal and Pell numbers will be stated.
Paper Structure (14 sections, 39 theorems, 157 equations, 1 table)

This paper contains 14 sections, 39 theorems, 157 equations, 1 table.

Table of Contents

  1. Motivation and Preliminaries
  2. General convolution identities with Fibonacci $m$-step numbers
  3. Mixed convolution
  4. Mixed convolutions with the "switch effect"
  5. Three other general identities
  6. Convolutions of multiple sequences
  7. A Pell-Fibonacci relation
  8. Convolution of three Fibonacci $m$-step sequences
  9. Convolutions of four Fibonacci $m$-step sequences
  10. Concluding results and comments
  11. Higher order convolutions
  12. Some general cases of convolution of two sequences
  13. Open problems
  14. Conclusion

Key Result

Theorem 2.1

For all $n\geq m$ we have

Theorems & Definitions (81)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Example 2.5
  • Example 2.6
  • Corollary 2.7
  • ...and 71 more