A note on a transform to self-inverse sequences

The sequences which are fixed by the binomial transform are called self-inverse sequences. In this paper, an identity satisfied by Fibonacci numbers is modified to provide a transform which maps a specific subset of sequences to self-inverse sequences bijectively. The image of some classes of sequences under this transform are explicitly found which provides a new formulation and a class of examples of self-inverse sequences. A criterion for the solutions of some difference equations to be self-inverse is also given.

___

  • [1] M. Berstein and N.J.A. Sloane, Some canonical sequences of integers, Linear Algebra Appl. 226-228, 57–72, 1995.
  • [2] K. Boyadzhiev, Binomial Transform and The Backward Difference, Adv. Appl. Discrete Math. 13 (1), 43–63, 2014.
  • [3] H. Prodinger, Some Information about the Binomial Transform, Fibonacci Quart. 32, 412–415, 1994.
  • [4] Z. Sun, Invariant sequences under binomial transformation, Fibonacci Quart. 39 (4), 324–333, 2001.
  • [5] R. Taurosa and S. Mattarei, Congruences of Multiple Sums Involving Sequences Invariant Under the Binomial Transform, J. Integer Seq. 13 (5), Article 10.5.1, 2010.
  • [6] Y. Wang, Self-inverse sequences related to a binomial inverse pair, Fibonacci Quart. 43 (1), 46–52, 2005.