Emrah KILIÇ, Yücel Türker ULUTAŞ, Neşe ÖMÜR

Sums of products of the terms of the generalized Lucas sequence {$V _{kn}$}

In this study we consider the generalized Lucas sequence {$V _{kn}$} with indices in arithmetic progression. We also compute the sums of products of the terms of the Lucas sequence {$V _{kn}$} for positive odd integers k.

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[1] Carlitz, L. Generating function for powers of a certain sequences of numbers, Duke Math. J. 29, 521–537, 1962.
[2] Dujella, A. A bijective proof of Riordan’s theorem on powers of Fibonacci numbers, Discrete Math. 199, 217–220, 1999.
[3] Golomb, S.W. Problem 4720, Amer. Math. Monthly 64, 49, 1957.
[4] Hoggatt Jr., V.E. Fibonacci numbers and generalized binomial coefficients, The Fibonacci Quarterly 5, 383–400, 1967.
[5] Horadam, A.F. Generating functions for powers of a certain generalized sequence of num- bers, Duke Math. J. 32, 437–446, 1965.
[6] Kılıç, E. and Stanica, P. Factorizations and representations of second linear recurrences with indices in arithmetic progressions, Bol. Mex. Math. Soc. 15 (1), 23–36, 2009.
[7] Kılıç, E. and Stanica, P. Factorizations and representations of binary polynomial recurrences by matrix methods, Rocky Mount. J. Math., in press.
[8] Riordan, J. Generating functions for powers of Fibonacci numbers, Duke Math. J. 29, 5–12, 1962.
[9] Riordan, J. Combinatorial Identities (J. Wiley, New York, 1968).
[10] Riordan, J. Inverse relations and combinatorial identities, Amer. Math. Monthly 71 (5), 485–498, 1964.
[11] Seibert, J. and Trojovsky, P. On sums of certain products of Lucas numbers, The Fibonacci Quarterly 44, 172–180, 2006.
[12] Seibert, J and Trojovsky, P. On multiple sums of products of Lucas numbers, J. Integer Seq. 10, 1–17, 2007.
[13] Shannon, A.G. A Method of Carlitz applied to the kth power generating function for Fi- bonacci numbers, The Fibonacci Quarterly 12, 293–299, 1974.
[14] Stanica, P. Generating function, weighted and non-weighted sums for powers of second-order recurrence sequences, The Fibonacci Quarterly 41 (4), 321–333, 2003.
[15] Vajda, S. Fibonacci and Lucas numbers and the Golden Section (Halsted Press, Brisbane, 1989).