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The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums

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  • Kılıç, Emrah

Abstract

The theory of generalized Pell p-numbers was introduced by Stakhov and then have been studied by several authors. In this paper, we consider the usual Pell numbers and as similar to the Fibonacci p-numbers, we give fair generalization of the Pell numbers, which we call the generalized Pell (p,i)-numbers for 0⩽i⩽p. First we give relationships between the generalized Pell (p,i)-numbers and give the generating matrices for these numbers. Also we derive the generalized Binet formulas, sums, combinatorial representations and generating function of the generalized Pell p-numbers. Also using matrix methods, we derive an explicit formula for the sums of the generalized Fibonacci p-numbers. Finally, we derive relationships between generalized Pell (p,i)-numbers and their sums and permanents of certain matrices.

Suggested Citation

  • Kılıç, Emrah, 2009. "The generalized Pell (p,i)-numbers and their Binet formulas, combinatorial representations, sums," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 2047-2063.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:4:p:2047-2063
    DOI: 10.1016/j.chaos.2007.09.081
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    References listed on IDEAS

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    1. Stakhov, Alexey & Rozin, Boris, 2006. "The continuous functions for the Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 1014-1025.
    2. El Naschie, M.S., 2006. "Elementary number theory in superstrings, loop quantum mechanics, twistors and E-infinity high energy physics," Chaos, Solitons & Fractals, Elsevier, vol. 27(2), pages 297-330.
    3. Stakhov, A.P., 2006. "Fibonacci matrices, a generalization of the “Cassini formula”, and a new coding theory," Chaos, Solitons & Fractals, Elsevier, vol. 30(1), pages 56-66.
    4. Stakhov, Alexey & Rozin, Boris, 2005. "The Golden Shofar," Chaos, Solitons & Fractals, Elsevier, vol. 26(3), pages 677-684.
    5. El Naschie, M.S., 2007. "On the topological ground state of E-infinity spacetime and the super string connection," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 468-470.
    6. Stakhov, Alexey & Rozin, Boris, 2006. "Theory of Binet formulas for Fibonacci and Lucas p-numbers," Chaos, Solitons & Fractals, Elsevier, vol. 27(5), pages 1162-1177.
    7. El Naschie, M.S., 2007. "From symmetry to particles," Chaos, Solitons & Fractals, Elsevier, vol. 32(2), pages 427-430.
    8. El Naschie, M.S., 2006. "On two new fuzzy Kähler manifolds, Klein modular space and ’t Hooft holographic principles," Chaos, Solitons & Fractals, Elsevier, vol. 29(4), pages 876-881.
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