FURTHER PROPERTIES OF GENERALIZED BINOMIAL COEFFICIENT k-extensions R. L. Ollerton University of Western Sydney, Penrith Capus DC1797, Australia A. G. Shannon KvB Institute of Technology, North Sydney 060, and Warrane College, The University of NSW, Kensington 1465, Australia (Subitted July 00) 1. INTRODUCTION Ollerton and Shannon [3] have developed k-extensions of the generalized binoial coefficients, written ( n with n,, 0 and 0 k abc 7, by considering the ways in which obects can be placed in n cells, each of which holds a axiu of obects, according to certain rules indicated by the binary variables a, b, c (where 1 Yes, 0 No). These correspond to: a. whether epty cells give new arrangeents (if not then epty cells ay be placed at the end of the arrangeent), b. whether the order of obects across the cells is free (if not then each cell ust contain obects all of which are less than those in the next non-blank cell to the right), and c. whether the order of obects within the cells gives new arrangeents (if not then they ay be listed in increasing order). k-extensions thus have very concrete cobinatorial interpretations and are a natural developent of the usual generalized binoial coefficients described by Bondarenko [1]. These latter have k 4 (whence 1 gives the ordinary binoial coefficients). A unifying recurrence relation is given by ( n il a ( n 1 Cib i! c i (1.1) for 0 k abc 7, n & > 0 and 0, with boundary conditions ( n 0 { n < 0 or < 0 n 0 & > 0 1 n 0 & 0. For each k and, arrays can be fored with rows given by n and coluns given by. Table 1 shows the 0-extension for. A nuber of properties of the k-extensions were developed in [3], including row and diagonal su recurrence relationships as well as the generating function: 0! b ( n x { T k (x)n+1 1 T k (x) 1 ( n + 1 for T k (x) 1 0 to 3 T k (x) n k 4 to 7 (1.) 14
where T k (x) il a i!c b x i. In the following, we develop further properties of these extensions by use of the generating function. Diagonal sus of the arrays produced by the k-extensions ay be considered to be extensions of the corresponding generalized Fibonacci seuences. We also deterine the diagonal su generating functions and describe soe of their properties.. FURTHER k-extension PROPERTIES Let g(k, n, ; x) n 0! b( n x. i. Differentiating (1.) with respect to x leads to g (k, n, ; x) 1! b ( n x 1 nt k (x)n+1 (n+1)t k (x)n +1 T k (T k(x) 1) ( n(n+1) T k (x) for T k (x) 1 (x 0 to 3 ) nt k (x) n 1 T k (x 4 to 7, noting that T k (x) i1 i!c b ix i 1. For exaple, setting x 1 and b c (k 0, 3, 4, 7) gives 1! b ( n T k (1) + a, T k (1) ( + 1)/ and { n n+1 (n+1) n +1 ( 1) ( + 1)/ ( ) n(n+1) for 1 k 0, 3 n( + 1) n / k 4, 7 which is well-known for k 4. For k 0 and in particular, 1 which ay be verified using Table 1. ( n 3(1 + (n 1) n ) ii. Recurrence relations in ters of ay also be developed fro (1.). For k 4 to 7 and 1, g(k, n, ; x) T k (x) n (T k 1(x) +! c b x ) n C n T 1(x (! c b x ) n 0 C n (! c b x ) n g(k,, 1; x) 0 15
( 0 ) n with g(k, n, 0; x) T0 k (x) n i0 i!c b x i 1. Setting x 1 gives recurrence relations for (weighted) row sus in ters of row sus of arrays with saller values of. Alternatingsign row su recurrence relations are found siilarly by setting x 1. Individual ters ay be explored further by euating powers of x. iii. The cases k 0 to 3 ay be related to k 4 to 7 respectively in the following anner. For k 0 to 3, T k (x) T k+4 (x) 1 and g(k, n, ; x) T k (x) n+1 1 T k (x) 1 T k (x) 0 (T k+4 0 0 h0 (x) 1) C h T k+4 (x) h ( 1) h g(k + 4, h, ; x) h0 C h ( 1) h. (.1) Once again, setting x ±1 provides relationships between the (weighted) row sus of the respective values of k. Individual coefficients ay be related by euating powers of x. For instance, it ay be verified fro Table 1 that ( n 0 n) F n+1. (Indeed, Table 1 indicates that there are F +1 ways of placing obects in n cells, where each cell holds at ost obects, according to the rules indicated by (a, b, c) (0, 0, 0).) For k 0 and, differentiating (.1) n ties with respect to x and then setting x 0 gives h so that g (n) (0, n, ; 0) n!f n+1 and g (n) (4, h, ; 0) { 0 h < n/ n! ( h n) 4 h n/ F n+1 h n+1 ( ) 4 h n C h ( 1) h. h This relates the Fibonacci seuence to the generalized binoial coefficients of order s +1 3 as described by Bondarenko. For exaple, when n 4, F 5 5 and 4 h ( ) 4 h 4 4 C h ( 1) h 1(C C 3 + C) 4 + 6(C3 3 C3) 4 + 19C4 4 5. h Relationships for other values of k and ay be explored in a siilar anner. 16
3. EXTENDED FIBONACCI SEQUENCE GENERATING FUNCTIONS Let d(k, n, ; x) 0 ( n/(+1) n ( n! b x! b x for n 0 (3.1) and d(k, n, ; x) 0 otherwise. This extends the concept of the usual generalized Fibonacci seuence (as defined for n 0). Substituting (1.1) for n, > 0 gives d(k, n, ; x) ( n + 0 1 a + 1 a +! b 1 i1 a i1 a i! c n i1 a 0 0 ( n 1 Cib i! c x i ( n 1 Cib! b x i n ( n 1 i! c b x i ( i)! b i 0 x i 1 a + 1 a + i1 a i1 a n i i! c b x i i n i 1 i! c b x i 0 ( n i 1! b x ( n i 1! b x (changing the suation index to i then reverting to ) (using the boundary conditions) 1 a + i! c b x i d(k, n i 1, ; x) (3.) i1 a with d(k, 0, ; x) 0 0! b( ) 0 k x ( 0 k 0) 1. 1 i. Diagonal su forulae, obtained here by setting x 1, were given in [3]. ii. Differentiating (3.) with respect to x leads to a diagonal su (further) weighted by colun nuber, d (k, n, ; x) 1 i1 a ( n! b x 1 i! c b ( ix i 1 d(k, n i 1, ; x) + x i d (k, n i 1, ; x) ). (3.3) 17
Substituting x 1 and k 4 in (3.1) leads to the generalized Fibonacci recurrence relation of arbitrary order with d(4, n 1, ; 1) U n defined by U n i0 U n i 1 for n > 1 and U n 0 for n 0,, 1, U 1 1. (3.3) then gives ( ) 4 n n i 1 ( ) 4 n i 1 iu n i +. 1 For exaple, 1 iplies U n F n and ( n ) 4 1 Cn which leads to 1 n 1 C n F n 1 + 1 i0 n C n 1 + 1 1 C n F n 1 + n 1 ( C n 1 Upper suation liits ay be reduced as in (3.1). When n 5 this gives C 4 1 + C 3 F 4 + C 3 1 + C 1 + C 10. ) + C n. Forulae for the other extensions ay be explored in a siilar anner. Substituting x 1 and k 0 in (3.1) leads to the dying rabbit seuence recurrence relations [] with d(0, n 1, ; 1) V n defined by V n 1 + i1 V n i 1 V n 1 + V n V n for n > 1 and V n 0 for n 0,, 1, V 1 1. (3.3) then gives ( n 1 iv n i + i1 n i 1 1 ( n i 1. For exaple, gives V n {0, 1, 1,, 3, 4, 6, 8, 11, 15,... } for n 0. Values of ( n are given in Table 1. This leads to ( n 1 V n 1 + V n + n ( n 1 + n 3 ( n 3 Upper suation liits ay also be reduced as in (3.1). When n 5, this gives 1 1 + + 3 3 + + (1 1 + 1 ) + 1 1 11. iii. Alternating-sign weighted diagonal sus can be investigated by setting x 1. 4. CONCLUSION The k-extensions of the generalized binoial coefficients provide a wealth of interesting relationships. Iportantly, these relationships ay also be interpreted in ters of the underlying cobinatorial eanings of the k-extensions. Further research as outlined by Ollerton 1. 18
and Shannon [3] should prove fruitful. 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 3 4 3 1 0 0 0 0 0 0 0 0 0 0 1 1 3 5 7 7 4 1 0 0 0 0 0 0 0 0 1 1 3 5 8 1 14 11 5 1 0 0 0 0 0 0 1 1 3 5 8 13 0 6 5 16 6 1 0 0 0 0 1 1 3 5 8 13 1 33 46 51 41 7 1 0 0 1 1 3 5 8 13 1 34 54 79 97 9 63 9 8 1 Table 1. The ( n 0 ) array for n 0 to 8 (n 0 gives rows, 0 gives coluns). REFERENCES [1] B.A. Bondarenko. Generalized Pascal Triangles and Pyraids. Translated by R.C. Bollinger. Santa Clara, California: The Fibonacci Association, 1993. [] F. Dubeau. The Rabbit Proble Revisited. The Fibonacci Quarterly 31.3 (1993): 68-74. [3] R.L. Ollerton & A.G. Shannon. Extensions of Generalized Binoial Coefficients. Applications of Fibonacci Nubers 9 (004): 187-199. AMS Classification Nubers: 11B39, 11B65, 15A36 19