Sitzungsber. Abt. II (2002) 211: 143–154 Some Relations Between Generalized Fibonacci and Catalan Numbers By Johann Cigler (Vorgelegt in der Sitzung der math.-nat. Klasse am 10. Oktober 2002 durch das w. M. Johann Cigler) Abstract In a recent paper Aleksandar Cvetkovicc, Predrag Rajkovic and Milosˇ Ivkovic proved that for the Catalan numbers Cn the Hankel determinants of the sequence Cn þ Cnþ1 are Fibonacci numbers. Their proof depends on special properties of the cor- responding orthogonal polynomials. In this paper we give a generalization of their result by other methods in order to give more insight into the situation. Mathematics Subject Classifications: 05A10, 05A19, 05A30, 11Bxx, 15A15. Key words: Catalan number, Fibonacci number, Hankel determinant, q-analogue, lattice path, LU-factorization. 1. A Survey of Known Results P À Á nÀ1 nÀ1Àk The classical Fibonacci polynomials Fnðx; sÞ¼ k¼0 k nÀ1À2k k  x Às Áare intimately related to the Catalan numbers 1 2n Cn ¼ nþ1 n : The Fibonacci polynomials Fnðx; 1Þ, n > 0, are a basis of the vector space of polynomials. If we define the linear 2nþ1 functional L by LðFnþ1Þ¼n;0 then we get Lðx Þ¼0and 2n n Lðx Þ¼ðÀ1Þ Cn. We will now sketch how this fact can be generalized. 144 J. Cigler 1 Given any sequence t ¼ðtðnÞÞn¼0 of positive real numbers we define the t-Fibonacci polynomials (cf. [5]) by Fnðx; tÞ¼xFnÀ1ðx; tÞþtðn À 3ÞFnÀ2ðx; tÞð1:1Þ with initial conditions F0ðx; tÞ¼0, F1ðx; tÞ¼1. If s is a real or complex number and tðnÞ¼s for all n 2 N this reduces to the classical Fibonacci polynomials Fnðx; sÞ introduced above. The first terms are 2 3 F2ðx; tÞ¼x; F3ðx; tÞ¼x þ tð0Þ; F4ðx; tÞ¼x þ xtð0Þþxtð1Þ; ... We state for later purposes the recurrences for the subsequences with even or odd indices. 2 F2nðx; tÞ¼ðx þ tð2n À 4Þþtð2n À 3ÞÞF2nÀ2ðx; tÞ À tð2n À 4Þtð2n À 3ÞF2nÀ4ðx; tÞð1:2Þ and 2 F2nþ1ðx; tÞ¼ðx þ tð2n À 3Þþtð2n À 2ÞÞF2nÀ1ðx; tÞ À tð2n À 3Þtð2n À 4ÞF2nÀ3ðx; tÞ: ð1:3Þ The polynomials Fnðx; tÞ, n > 0, are a basis of the vector space P of all polynomials in x. We can therefore define a linear functional L on P by LðFnÞ¼n;1: ð1:4Þ ^ Fnðx;tÞ Let Fnðx; tÞ¼tð0Þtð1ÞtðnÀ2Þ : Then we have xF^n ¼ tðn À 1ÞF^nþ1 À F^nÀ1: nþk d 2 e n Define now the numbers an;k ¼ðÀ1Þ Lðx F^kþ1Þ, where dxe de- notes the least integer greater than or equal to x. They satisfy a0;k ¼ 0;k an;k ¼ anÀ1;kÀ1 þ tðkÞanÀ1;kþ1 ð1:5Þ where, an;k ¼ 0ifk < 0. They have an obvious combinatorial interpretation. Consider all nonnegative lattice paths in R2 which start in ð0; 0Þ with upward steps ð1; 1Þ und downward steps ð1; À1Þ. We associate to each upward step ending on the height k the weight 1 and to each downward step ending on the height k the weight tðkÞ. The weight of the path is the product of the weights of all steps of the path. Then an;k is the weight of all nonnegative lattice paths from ð0; 0Þ to ðn; kÞ. Some Relations Between Generalized Fibonacci and Catalan Numbers 145 It is clear that a2nþ1;0 ¼ 0. If we set a2n;0 ¼ CnðtÞ, then CnðtÞ, which we call a t-CatalanÀ numberÁ (cf. [4]), is an analogue of the 1 2n Catalan number Cn ¼ nþ1 n , because it is well known that the number of such paths equals Cn. It is easy to give a recurrence for these t-Catalan numbers. To this end decompose each lattice path from ð0; 0Þ to ð2n; 0Þ into the first path which returns to the x-axis and the rest path. The first path goes from ð0; 0Þ to ð2k þ 2; 0Þ,0 k n À 1, and consists of a rising segment followed by a path from ð0; 0Þ to ð2k; 0Þ (but one level higher) and a falling segment. Thus XnÀ1 CnðtÞ¼tð0Þ CkðEtÞCnÀkÀ1ðtÞ; C0ðtÞ¼1: ð1:6Þ k¼0 Here Et denotes the shifted sequence Et ¼ðtð1Þ; tð2Þ; ...Þ. This is an analogue of the recursion XnÀ1 Cn ¼ CkCnÀkÀ1; C0 ¼ 1; k¼0 for the classical Catalan numbers. Since 2nþ1 2n n Lðx Þ¼0 and Lðx Þ¼ðÀ1Þ CnðtÞ we get Theorem 1. Let L be the linear functional on the vector space P of all polynomials defined by (1.4). This can be characterized by 2n n 2nþ1 Lðx Þ¼ðÀ1Þ CnðtÞ; Lðx Þ¼0 ð1:7Þ for all n 2 N. This theorem (cf. [5]) gives a connection between t-Fibonacci polynomials and t-Catalan numbers. The purpose of this paper is to show another connection between these numbers. First we propose an elementary method using LU-factorization (cf. e.g. [8], where also other methods for treating Hankel determinants are listed). To this end we generalize two well-known triangular Catalan matrices (cf. e.g. [1, 4, 9]): Let cðn; kÞ¼a2n;2k: ð1:8Þ This gives cðn; 0Þ¼CnðtÞ. 146 J. Cigler Then it is clear that the following recursion holds: cð0; kÞ¼½k ¼ 0 cðn; 0Þ¼tð0Þcðn À 1; 0Þþtð0Þtð1Þcðn À 1; 1Þ; n > 0; cðn; kÞ¼cðn À 1; k À 1Þþðtð2k À 1Þþtð2kÞÞcðn À 1; kÞþ þ tð2kÞtð2k þ 1Þcðn À 1; k þ 1Þ; n > 0; k > 0: ð1:9Þ We may interpret cðn; kÞ as the weight of another sort of nonnegative lattice paths from ð0; 0Þ to ðn; kÞ with upward, downward and horizontal steps, where each upward step has weight 1, each downward step ending at height k has weight tð2kÞtð2k þ 1Þ and each horizontal step in height k has weight tð2k À 1Þþtð2kÞ. Setting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ccðn; kÞ¼ tð0Þtð1Þtð2k À 1Þcðn; kÞð1:10Þ we get another weight on these lattice paths which is symmetric, i.e. upward steps and downward steps between the same heights have the same weight. Furthermore we have ccðn; 0Þ¼cðn; 0Þ¼CnðtÞ. They satisfy ccð0; kÞ¼½k ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ccðn; 0Þ¼tð0Þccðn À 1; 0Þþ tð0Þtð1Þcðn À 1; 1Þ; n > 0; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ccðn; kÞ¼ tð2k À 2Þtð2k À 1Þccðn À 1; k À 1Þþ þðtð2k À 1Þþtð2kÞÞccðn À 1; kÞþ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ tð2kÞtð2k þ 1Þcðn À 1; k þ 1Þ; n > 0; k > 0: ð1:11Þ If we decompose a lattice path from ð0; 0Þ to ðm þ n; 0Þ into a path from ð0; 0Þ to ðm; kÞ and a second path from ðm; kÞ to ðm þ n; 0Þ, then the weight of the second path is identical with ccðn; kÞ because of the symmetry. This gives the identity X ccðm; kÞccðn; kÞ¼ccðm þ n; 0Þð1:12Þ k0 or equivalently X cðm; kÞcðn; kÞtð0Þtð1Þtð2k À 1Þ¼cðm þ n; 0Þ¼CmþnðtÞ: k0 This may be interpreted in the following form: Consider the triangular matrices nÀ1 Pn ¼ðccði; jÞÞi; j¼0: ð1:13Þ Some Relations Between Generalized Fibonacci and Catalan Numbers 147 Then t nÀ1 PnPn ¼ðCiþjðtÞÞi; j¼0: ð1:14Þ This leads immediately to the determinant of the Hankel matrix YnÀ1 nÀ1 t det ðCiþjðtÞÞi; j¼0 ¼ detPnPn ¼ tð0Þtð1Þtð2k À 1Þð1:15Þ k¼0 For the second Catalan matrix let dðn; kÞ¼a2nþ1;2kþ1: ð1:16Þ Then dðn; 0Þ¼a2nþ1;1 ¼ a2nþ2;0 ¼ Cnþ1ðtÞ: ð1:17Þ The recursion takes now the following form: dð0; kÞ¼tð0Þ½k ¼ 0 dðn; 0Þ¼ðtð0Þþtð1ÞÞdðn À 1; 0Þþtð1Þtð2Þdðn À 1; 1Þ; n > 0; dðn; kÞ¼dðn À 1; k À 1Þþðtð2kÞþtð2k þ 1ÞÞdðn À 1; kÞþ þ tð2k þ 1Þtð2k þ 2Þdðn À 1; k þ 1Þ; n > 0; k > 0: ð1:18Þ Setting pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tð1Þtð2kÞ ddðn; kÞ¼ dðn; kÞð1:19Þ tð0Þ we get another weight on these lattice paths which is symmetric and satisfies ddð0; kÞ¼½k ¼ 0. This gives the identity X ddðm; kÞddðn; kÞ¼ddðm þ n; 0Þ¼Cmþnþ1ðtÞð1:20Þ k0 or equivalently X dðm; kÞdðn; kÞtð1Þtð2kÞ¼tð0Þdðm þ n; 0Þ¼tð0ÞCmþnþ1ðtÞ: k0 This result may be interpreted in the following form: Consider the triangular matrices nÀ1 Qn ¼ðddði; jÞÞi; j¼0: ð1:21Þ 148 J. Cigler Then t nÀ1 QnQn ¼ðCiþjþ1ðtÞÞi; j¼0 ð1:22Þ and we get the determinant of the second Hankel matrix Yn nÀ1 t detðCiþjþ1ðtÞÞi;j¼0 ¼detQnQn ¼ tð0Þtð1Þtð2Þtð2kÀ2Þ: ð1:23Þ k¼0 Remark. This implies that t t det QnQn ¼ tð0Þtð2ÞÁÁtð2n À 2Þdet PnPn: ð1:24Þ The most important special cases are the following: 1) tðnÞ¼qns: The Carlitz q-Fibonacci polynomials and the Carlitz q-Catalan numbers. Here we get ([2–4]) n PnÀ1 n n YnÀ1 2 ðnÀ1Þnð4nÀ5Þ 2 2 i À 2 tð0Þtð1ÞÁÁtð2k À 1Þ¼q 6 s 2 ¼ q i¼0 2 s 2 k¼0 and PnÀ1 n Yn 2 i2þ nðnÀ1Þð4nþ1Þ n2 n2 tð0Þtð1ÞÁÁtð2k À 2Þ¼q 6 s ¼ q i¼0 2 s : k¼0 It should be noted that in this case k XnÀ1 n À k À 1 2 F ðx; s; qÞ¼ q 2 skxnÀ2kÀ1: n k k¼0 2) Another interesting special case arises for tð2kÞ¼rqk; tð2k þ 1Þ¼sqk; where r and s are positive real numbers. The corresponding Fibonacci polynomials are 0; 1; x; x2 þ r; x3 þ rx þ sx; x4 þ rx2 þ qrx2 þ sx2 þ qr2; ...: The corresponding Catalan numbers Cnðq; r; sÞ (the Polya- Gessel q-Catalan numbers) are 1; r; r2 þ rs; r3 þ 2r2s þ qr2s þ rs2; ...: Some Relations Between Generalized Fibonacci and Catalan Numbers 149 They satisfy the recurrence (cf. [4, 7]) XnÀ1 k Cnþ1ðq; r; sÞ¼rCnðq; r; sÞþs q Ckðq; r; sÞCnÀkðq; r; sÞ: k¼0 In this case we have ([4]) n n YnÀ1 2 tð0Þtð1Þtð2k À 1Þ¼q 3 ðrsÞ 2 k¼0 and PnÀ1 n þ 1 n Yn k2 tð0Þtð1Þtð2k À 2Þ¼q k¼0 r 2 s 2 : k¼0 2.