| 释义 |
Generalized Fibonacci NumberA generalization of the Fibonacci Numbers defined by  and the Recurrence Relation
 | (1) |
 up and 1 right. The case  equals the usual Fibonacci Number. These numberssatisfy the identities
 | (2) |
 | (3) |
 | (4) |
 | (5) |
 ,
 | (6) |
Horadam (1965) defined the generalized Fibonacci numbers  as  , where  ,  ,  , and  are Integers,  ,  , and  for  . They satisfy the identities
 | (7) |
 | (8) |
 | (9) |
 | |  | (10) |
where
The final above result is due to Morgado (1987) and is called the Morgado Identity.
Another generalization of the Fibonacci numbers is denoted  . Given  and  , define the generalized Fibonaccinumber by  for  ,
 | (13) |
 | (14) |
 | (15) |
References
Bicknell, M. ``A Primer for the Fibonacci Numbers, Part VIII: Sequences of Sums from Pascal's Triangle.'' Fib. Quart. 9, 74-81, 1971.Bicknell-Johnson, M. and Spears, C. P. ``Classes of Identities for the Generalized Fibonacci Numbers  for Matrices with Constant Valued Determinants.'' Fib. Quart. 34, 121-128, 1996. Dujella, A. ``Generalized Fibonacci Numbers and the Problem of Diophantus.'' Fib. Quart. 34, 164-175, 1996. Horadam, A. F. ``Generating Functions for Powers of a Certain Generalized Sequence of Numbers.'' Duke Math. J. 32, 437-446, 1965. Horadam, A. F. ``Generalization of a Result of Morgado.'' Portugaliae Math. 44, 131-136, 1987. Horadam, A. F. and Shannon, A. G. ``Generalization of Identities of Catalan and Others.'' Portugaliae Math. 44, 137-148, 1987. Morgado, J. ``Note on Some Results of A. F. Horadam and A. G. Shannon Concerning a Catalan's Identity on Fibonacci Numbers.'' Portugaliae Math. 44, 243-252, 1987. |
