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A multi-parameter generalization of the symmetric algorithm

José L. Ramírez and Mark Shattuck
From the journal Mathematica Slovaca


The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

  1. Communicated by Stanislav Jakubec


[1] Andrews, G. E.: The Theory of Partitions, Cambridge University Press, 1998. Search in Google Scholar

[2] Bahşi, M.—Mező, I.—Solak, S.: A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers, Ann. Math. Inform. 43 (2014), 19–27. Search in Google Scholar

[3] Belbachir, H.—Belkhir, A.: Combinatorial expressions involving Fibonacci, hyperfibonacci, and incomplete Fibonacci numbers, J. Integer Seq. 17 (2014), Art. 14.4.3. Search in Google Scholar

[4] Belbachir, H.—Benmezai, A.: An alternative approach to Cigler’s q-Lucas polynomials, Appl. Math. Comput. 226 (2014), 691–698. Search in Google Scholar

[5] Benjamin, A. T.—Eustis, A. K.—Shattuck, M. A.: Compression theorems for periodic tilings and consequences, J. Integer Seq. 12 (2009), Art. 09.6.3. Search in Google Scholar

[6] Benjamin, A. T.—Quinn, J. J.: Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, Washington DC, 2003. Search in Google Scholar

[7] Briggs, K.—Little, D.—Sellers, J.: Combinatorial proofs of various q-Pell identities via tilings, Ann. Comb. 14 (2010), 407–418. Search in Google Scholar

[8] Carlitz, L.: Fibonacci notes 3, q-Fibonacci numbers, Fibonacci Quart. 12 (1974), 317–322. Search in Google Scholar

[9] Carlitz, L.: Fibonacci notes 4, q-Fibonacci polynomials, Fibonacci Quart. 13 (1975), 97–102. Search in Google Scholar

[10] Cigler, J.: A new class of q-Fibonacci polynomials, Electron. J. Combin. 10 (2003), #R19. Search in Google Scholar

[11] Cigler, J.: Some algebraic aspects of Morse code sequences, Discrete Math. Theor. Comput. Sci. 6 (2003), 55–68. Search in Google Scholar

[12] Clarke, R.—Han, G.—N.—Zeng, J.: A combinatorial interpretation of the Seidel generation of q-derangement numbers, Ann. Comb. 4 (1997), 313–327. Search in Google Scholar

[13] Corcino, R.: On p,q-binomial coefficients, Integers 8 (2008), #A29. Search in Google Scholar

[14] Dil, A.—Mező, I.: A symmetric algorithm for hyperharmonic and Fibonaci numbers, Appl. Math. Comput. 206 (2008), 942–951. Search in Google Scholar

[15] Dumont, D.—Randrianarivony, A.: Dérangements et nombres de Genocchi, Discrete Math. 132 (1994), 37–49. Search in Google Scholar

[16] Firengiz, M. C.—Tasci, D.—Tuglu, N.: Some identities for Fibonacci and incomplete Fibonacci p-numbers via the symmetric matrix method, J. Integer Seq. 17 (2014), Art. 14.2.5. Search in Google Scholar

[17] Garrett, K.: Weighted tilings and q-Fibonacci numbers, preprint. Search in Google Scholar

[18] Little, D.—Sellers, J.: A tiling approach to eight identities of Rogers, European J. Combin. 31 (2010), 694–709. Search in Google Scholar

[19] Mező, I.—Ramirez, J.: A q-symmetric algorithm and its applications to some combinatorial sequences, Online J. Anal. Comb. 12 (2017), Art. 7. Search in Google Scholar

[20] Shattuck, M.—Wagner, C.: A new statistic on linear and circular r-mino arrangements, Electron. J. Combin. 13 (2006), #R42. Search in Google Scholar

[21] Shattuck, M.—Wagner, C.: Some generalized Fibonacci polynomials, J. Integer Seq. 10 (2007), Art. 7.5.3. Search in Google Scholar

[22] Stabel, E. C.: A combinatorial proof of an identity of Ramanujan using tilings, Bull. Braz. Math. Soc. 42 (2011), 203–212. Search in Google Scholar

[23] Stanley, R. P.: Enumerative Combinatorics, Vol. 1, Cambridge University Press, 1997. Search in Google Scholar

Received: 2016-07-01
Accepted: 2017-02-12
Published Online: 2018-08-06
Published in Print: 2018-08-28

© 2018 Mathematical Institute Slovak Academy of Sciences