# Linear recurrences associated to rays in Pascal’s triangle and combinatorial identities

• Hacène Belbachir , Takao Komatsu and László Szalay
From the journal Mathematica Slovaca

## Abstract

Our main purpose is to describe the recurrence relation associated to the sum of diagonal elements laying along a finite ray crossing Pascal’s triangle. We precise the generating function of the sequence of described sums. We also answer a question of Horadam posed in his paper [Chebyshev and Pell connections, Fibonacci Quart. 43 (2005), 108–121]. Further, using Morgan-Voyce sequence, we establish the nice identity $F_{n + 1} - iF_n = i^n \sum\limits_{k = 0}^n {(_{2k}^{n + k} )( - 2 - i)^k }$ of Fibonacci numbers, where i is the imaginary unit. Finally, connections to continued fractions, bivariate polynomials and finite differences are given.

[1] AMDEBERHAN, T.: A note on Fibonacci polynomials, Integers 10 (2010), 13–18. http://dx.doi.org/10.1515/integ.2010.00210.1515/integ.2010.002Search in Google Scholar

[2] ANDRÉ-JEANNIN, R.: A generalization of Morgan-Voyce polynomials, Fibonacci Quart. 32 (1994), 228–231. Search in Google Scholar

[3] BELBACHIR, H.— BENCHERIF, F.: Linear recurrent sequences and powers of a square matrix, Integers 6 (2006), A12, 17 pp. Search in Google Scholar

[4] BELBACHIR, H.— BENCHERIF, F.: On some properties of bivariate Fibonacci and Lucas polynomials, J. Integer Seq. 11 (2008), Article 08.2.6, 10 pp. Search in Google Scholar

[5] CHU, W.— VICENTI, V.: Generating functions and incomplete Fibonacci and Lucas polynomials, Boll. Unione Mat. Ital. Sez. B. Artic. Ric. Mat. (8) 6 (2003), 289–308. Search in Google Scholar

[6] GOULD, H. W.: Combinatorial Identities. A Standardized Set of Tables Listing 500 Binomial Coefficient Summations, Henry E. Gould, Morgantown, W. Va., 1972. Search in Google Scholar

[7] HORADAM, A. F.: Chebyshev and Fermat polynomials for diagonal functions, Fibonacci Quart. 17 (1979), 328–333. Search in Google Scholar

[8] HORADAM, A. F.: Quasi Morgan-Voyce polynomials and Pell convolutions. In: Applications of Fibonacci Numbers, Vol. 8, Rochester, NY, Kluwer Acad. Publ., Dordrecht, 1999, p. 998. Search in Google Scholar

[9] HORADAM A. F.: Chebyshev and Pell Connections, Fibonacci Quart. 43 (2005), 108–121. Search in Google Scholar

[10] JONES, W. B.— THRON, W. J.: Continued fractions. In: Analytic Theory and Applications, Encyclopedia of Mathematics and Its Applications, Vol. 11, Addison-Wesley, Reading, MA, 1980. Search in Google Scholar

[11] KOSHY, T.: Fibonacci and Lucas Numbers with Applications, John Wiley & Sons, New York, 2001. http://dx.doi.org/10.1002/978111803306710.1002/9781118033067Search in Google Scholar

[12] LUCAS, E.: Théorie des nombres, Ghautier-Villars, Paris, 1891. Search in Google Scholar

[13] MORGAN-VOYCE, A. M.: Ladder networks analysis using Fibonacci numbers, IEEE Trans. Circuits Theory 6 (1959), 321–322. http://dx.doi.org/10.1109/TCT.1959.108656410.1109/TCT.1959.1086564Search in Google Scholar

[14] VAN DER POORTEN, A. J.: Continued fraction expansions of values of the exponential function and related fun with continued fractions, Nieuw Arch. Wiskd. (5) 14 (1996), 221–230. Search in Google Scholar

[15] The Online Encyclopedia of Integer Sequences, http://www.research.att.com/njas/sequences. Search in Google Scholar

[16] SPIVEY, M. Z.: Combinatorial sums and finite differences, Discrete Math. 307 (2007), 3130–3146. http://dx.doi.org/10.1016/j.disc.2007.03.05210.1016/j.disc.2007.03.052Search in Google Scholar

[17] SPRUGNOLI, R.: Riordan rays and combinatorial sums, Discrete Math. 132 (1994), 267–290. http://dx.doi.org/10.1016/0012-365X(92)00570-H10.1016/0012-365X(92)00570-HSearch in Google Scholar

[18] SWAMY, M. N. S.: Properties of the polynomial defined by Morgan-Voyce, Fibonacci Quart. 4 (1966), 73–81. Search in Google Scholar

[19] SWAMY, M. N. S.— BHATTACHARYYA, B.: A study of recurrent ladders using the polynomials defined by Morgan-Voyce, IEEE Trans. Circuits Theory 14 (1967), 260–264. http://dx.doi.org/10.1109/TCT.1967.108270510.1109/TCT.1967.1082705Search in Google Scholar

Published Online: 2014-5-8
Published in Print: 2014-4-1