Bull. Math. Soc. Sci. Math. Roumanie Tome 55(103 No. 1, 01, 51 61 Some Generalized Fibonomial Sums related with the Gaussian q Binomial sums by Emrah Kilic, Iler Aus and Hideyui Ohtsua Abstract In this paper, we consider some generalized Fibonomial sums formulae and then prove them by using the Cauchy binomial theorem and q Zeilberger algorithm in Mathematica session. Key Words: Gaussian binomial coefficient, fibonomial coefficient, q binomial theorem, sums. 010 Mathematics Subject Classification: Primary 11B65; Secondary 05A10, 11B37. 1 Introduction For all real n and integer with 0, the Gaussian q binomial coefficients is defined by and as zero otherwise, where [ ] n : q (q; q n (q; q (q; q n (x; q n : (1 x(1 xq... (1 xq n 1. [ ] n q Thus far, the q identities have taen the interest of many authors. For a detailed information about the q identities, we may refer to [1, 6, 7] and its to the list of references. We recall some well nown identities related to the q identities: firstly Gauss identity is as follows : n ( 1 [ n ] q (1 q 1 1
5 Emrah Kilic, Iler Aus and Hideyui Ohtsua and the Cauchy binomial theorem is given by n [ ] n y q (+1/ q n ( 1 + yq. (1.1 Define the non-degenerate second order linear sequences {U n } and {V n } by, for n > 1 1 U n pu n 1 + U n, U 0 0, U 1 1, V n pv n 1 + V n, V 0, V 1 p. The Binet formulas of {U n } and {V n } are where α, β U n αn β n α β ( p ± p + 4 /. and V n α n + β n, For n 1 and any positive integer m, define the generalized Fibonomial coefficient by { } n U m U m... U nm : (U m U m... U m ( U m U m... U m(n i1 U m(n i+1 U mi with { n 0 } { } n n 1. When m 1, we obtain the usual Fibonomial coefficient, denoted by { } n. U For more details about the Fibonomial and generalized Fibonomial coefficients, see [, 3, 4, 9]. The lin between the generalized Fibonomial and Gaussian q binomial coefficients is { } n α m(n [ ] n with q α. By taing q β/α, the Binet formulae are reduced to the following forms: n 1 1 qn U n α 1 q and V n α n (1 + q n, where i 1 α q. All the identities we will derive hold for general q, and results about generalized Fibonacci and Lucas numbers come out as corollaries for the special choice of q. In the present study, we shall consider new generalized Fibonomial sums formulae and prove them by using the Cauchy binomial theorem and the q Zeilberger algorithm (for such computer algorithms, we may refer to [5].
Some Generalized Fibonomial Sums Formulas 53 Some Fibonomial Sums In this section, we consider some generalized Fibonomial sums and compute them by means of Gaussian q-binomial sums formulae. Lemma 1. For n 0, n+1 (i i a( 1 i (ii (iii n n+1 i m ± b( i mn n+1 n a(, i ± b(, i (n+1 ± a( 1 ± ( 1n i n+1 a(, where the sequences a ( and b ( satisfy a( a(n + 1 and b( b(n, respectively. Proof: Consider n+1 (i i a( 1 n+1 (i + i (n+1 a( 1 i n+1 a(. (ii Let C n;m ( : i mn ± i m ±. Then we have C n;m (+C n;m (n 0. Therefore, n C n;m (b( 1 n (C n;m ( + C n;m (n b( 0. This result together with some manipulations proves (ii. n+1 (iii i (n+1 + a( 1 n+1 1 + ( 1n i ( ( 1 (n+1 i + ( 1 (n+1 (n+1 i (n+1 a( n+1 a(. By a similar thin, it is seen that n+1 i (n+1 a( 1 ( 1n i n+1 a(, which completes the proof.
54 Emrah Kilic, Iler Aus and Hideyui Ohtsua Lemma. For positive integers n and m, n [ ] ( 1 m(n+1 ( n 1 n + 1 { ( 1 (n+1 q m ( ; n if m is odd, q m(n+1 ( ; n if m is even. Proof: If m is odd, n n+1 n+1 ( q m( n 1 ( ( n 1 [ ] n + 1 [ ] n + 1 ( +1 [ n + 1 i (i (n+1 q m(n+1 ] which, by (i in Lemma 1, gives us n+1 ( +1 [ ] n + 1 (i (n+1 q m(n+1 1 i n+1 (1 + i (1 + i (n+1 ( n 1 1 (1 + i(1 + i (n+1 (1 + i (n+1 j (1 + i (n+1 q mj j1 n i n i (n+1 (j + q mj j1 ( q m ( ; n. Thus the proof is complete for the case m is odd.
Some Generalized Fibonomial Sums Formulas 55 If m is even, n n+1 n+1 1 ( q m( n 1 ( n 1 [ ] n + 1 [ ] n + 1 ( +1 [ n + 1 q m(n+1 n+1 (1 + ( n 1 (1 + j (1 + q mj j1 j1 ( + j + q mj q mj (1 + j q m ( ; n. j1 ] So we have the conclusion. Theorem 1. For positive integers n and m, n { } n + 1 1 1 V m V m if m is odd, if m is even. Proof: The proof could be obtained from Lemma by taing q β/α. Lemma 3. For even m and any integer t, n ( 1 ( m +t ( n [ ] n 1 (( 1 m +t q m( 1 (1 + ( 1 +.
56 Emrah Kilic, Iler Aus and Hideyui Ohtsua Proof: Consider the RHS of the claimed identity, we write n n 1 j1 j1 ( 1 ( m +t ( n [ ] n ( +1 [ (( 1 m +t q m(n+1 n (1 + ( 1 m +t ( n 1 (1 + ( 1 m +t (j 1 (1 + ( 1 m +t q m(j 1 ] (( 1 m +t q m(j 1 ((j 1 + 1 +. Thus the proof is complete. Theorem. For even m and any integer t, n { } n ( 1 t ( ( 1 t V m( 1 +. 1 As a consequence of Theorem, we have the following Corollary: Corollary 1. For even m, n { } { n (i 1 n ( Vm( 1 + ( + Vm( 1 + }, 1 1 n { } { n (ii 1 n ( Vm( 1 + ( Vm( 1 + }. 1 1 1 Lemma 4. For any positive integers n and m, n i (mn m±1 ( n [ ] n 1 i n(mn±1 q mn ( ; n. Proof: Consider the RHS of the claimed identity and by using (ii in Lemma 1,
Some Generalized Fibonomial Sums Formulas 57 we get n n i (mn m±1 ( n [ ] n ( +1 ( [ ] i m i mn±1 q m(n+1 n n ( +1 ( [ ] i mn i mn±1 q m(n+1 n i mn 1 i mn n j1 i mn n j1 (1 + i mn±1 ( n 1 (1 + i mn±1 (j 1 i mn±1 q m(j 1 (1 + (j 1 i mn ±n q mn ( ; n. (1 + i mn±1 q m(j 1 So we have the claimed identity. Theorem 3. For any positive integers n and m, n i ± { n } n i ±n V m( 1. 1 Using Theorem 3, the following corollary can be obtained: Corollary. For any positive integers n and m, n { } n (i ( 1 cos nπ V m( 1, 1 n { } n (ii ( 1 sin nπ V m( 1. 1 1 Lemma 5. For any positive integers n and m, n+1 i (m( n 1±1 ( n 1 (1 ± ii ±n [ ] n + 1 ( 1 q m ( ( ; n n+1 q m ( ; n if m is odd, if m is even.
58 Emrah Kilic, Iler Aus and Hideyui Ohtsua Proof: If m is odd, then we write n+1 n+1 i m(n+1 ± ( n 1 i (n+1 ± ( n 1 [ ] n + 1 [ ] n + 1 which, by (iii in Lemma 1, gives us 1 ± ( 1n i 1 ± ( 1n i 1 ± ( 1n i n+1 n+1 (1 ± ( 1 n i (1 ± ( 1 n i ( +1 [ ] n + 1 q m(n+1 (1 + ( n 1 1 (1 + j (1 + q mj j1 ( + j + q mj j1 q mj (1 + j j1 (1 ± ( 1 n iq m ( ; n (1 ± ii ±n ( q m ( ; n. Thus the claim is seen for odd m.
Some Generalized Fibonomial Sums Formulas 59 If m is even, then we similarly obtain by (1.1, n+1 [ ] i m(n+1 ± ( n 1 n + 1 n+1 [ ] i ± ( n 1 n + 1 n+1 ( +1 ( i ±1 q m(n+1 [ ] n + 1 n+1 (1 + i ±1 ( n 1 (1 + i ±1 (1 + i ±1 j (1 + i ±1 q mj 1 (1 ± i j1 i ±1 (j + q mj (1 ± ii ±n j1 (1 ± ii ±n q m ( ; n. n j1 q mj (1 + j Theorem 4. For any positive integers n and m, n+1 { } n + 1 i ± (1 ± ii ±n 1 1 V m V m 1 if m is odd, if m is even. Using Theorem 4, we have the following Corollary: Corollary 3. For any positive integers n and m, n { } n + 1 Vm if m is odd, (i ( 1 ( 1 (n+1 1 V m if m is even. 1 n { } n + 1 Vm if m is odd, (ii ( 1 ( 1 (n 1 + 1 V m if m is even. Lemma 6. For positive integers n and m, n ( 1 ( n [ n ] ( 1 n q mn ( ; n ( ; q 4m n ( ; n. (.1
60 Emrah Kilic, Iler Aus and Hideyui Ohtsua Proof: We will show how to evaluate and prove such sums entirely mechanically. We used Zeilberger s own version [8], which is a Mathematica program. After loading the pacage qzeil.m in Mathematica, define SUM[n] : n [ ] ( 1 n ( n. (. If we run the RHS of (. in the program, then we obtain following first order recurrence relation: SUM[n] qm mn (1 + n (1 (n 1 1 n SUM[n 1]. Clearly the right hand side of equation (.1 satisfies the same recurrence relation (and the both sides of (.1 have the same initial conditions. Also, if we solve this recurrence relation, we obtain SUM[n] (-1n 1 mn ( q 3m ; n 1 (q 3m ; n 1 ( q 4m ; n 1 (q 4m. (.3 ; n 1 Thus, the right hand side of (.1 and (.3 are the same; it is no more necessary to distinguish the parity of m. Consequently, the proof is complete. Theorem 5. For positive integers n and m, References n ( 1 (m+1 { n } ( 1 (m+1n n 1 V m U m( 1 U m. [1] G. E. Andrews, The Theory of Partitions, Encyclopedia of mathematics and its applications, Vol., Addison-Wesley, Reading, MA, 1976. [] V.E. Hoggatt Jr., Fibonacci numbers and generalized binomial coefficients, Fibonacci Quart. 5 (1967 383 400. [3] E. Kilic, The generalized Fibonomial matrix, European J. Combin. 31 (1 (010 193-09. [4] E. Kilic and P. Stanica, Generating matrices of C -nomial coefficients and their spectra, Proc. International Conf. Fibonacci Numbers & Applic. 010, accepted. [5] M. Petovse, H.Wilf, and D.Zeilberger, A B, A K Peters, Wellesley, MA, 1996.
Some Generalized Fibonomial Sums Formulas 61 [6] H. Prodinger, Some applications of the q-rice formula, Random Structures and Algorithms 19 (3-4 (001 55 557. [7] A. Riese, A Mathematica q-analogue of Zeilberger s Algorithm for Proving q Hypergeometric Identities, Diploma Thesis, RISC, J. Kepler University, Linz, Austria, 1995. [8] A. Riese, http://www.risc.uni-linz.ac.at/research/combinat/software/ qzeil/index.php. [9] P. Trojovsy, Pavel On some identities for the Fibonomial coefficients via generating function, Discrete Appl. Math. 155 (15 (007 017 04. TOBB University of Economics and Technology Mathematics Department 06560 Sögütözü Anara Turey E-mail: eilic@etu.edu.tr Kırıale University Department of Mathematics 71450 Yahşihan Kırıale Turey E-mail: ileraus@u.edu.tr Bunyo University High School, 1191-7, Kami, Ageo-city, Saitama Pref., 36-0001, Japan E-mail: otsuahideyui@gmail.com