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SEVERAL CLOSED and DETERMINANTAL FORMS for CONVOLVED FIBONACCI NUMBERS Contents 1. Background and Motivations 1 2. Lemmas 2 3. M

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SEVERAL CLOSED and DETERMINANTAL FORMS for CONVOLVED FIBONACCI NUMBERS Contents 1. Background and Motivations 1 2. Lemmas 2 3. M SEVERAL CLOSED AND DETERMINANTAL FORMS FOR CONVOLVED FIBONACCI NUMBERS MUHAMMET CIHAT DAGLI˘ AND FENG QI Abstract. In the paper, with the aid of the Fa`adi Bruno formula, some properties of the Bell polynomials of the second kind, and a general derivative formula for a ratio of two differentiable functions, the authors establish several closed and determinantal forms for convolved Fibonacci numbers. Contents 1. Background and motivations 1 2. Lemmas 2 3. Main results 4 References 8 1. Background and motivations The Fibonacci numbers Fn are defined by the recurrence relation Fn = Fn−1 + Fn−2 for n ≥ 2 subject to initial values F0 = 0 and F1 = 1. All the Fibonacci numbers Fn for n ≥ 1 can be generated by 1 1 X = F tn = 1 + t + 2t2 + 3t3 + 5t4 + 8t5 + ··· 1 − t − t2 n+1 n=0 Numerous types of generalizations of these numbers have been offered in the liter- ature and they have a plenty of interesting properties and applications to almost every field of science and art. For more information, please refer to [11, 18, 32] and closely related references therein. For r 2 R, convolved Fibonacci numbers Fn(r) can be defined by 1 1 X = F (r)tn: (1) (1 − t − t2)r n+1 n=0 See [1, 2, 6, 14, 19] and closely related references therein. It is easy to see that Fn(1) = Fn for n ≥ 1. Convolved Fibonacci numbers Fn(r) for n 2 N and their generalizations have been considered in the papers [8, 10, 12, 14, 19, 37, 38, 39] 2010 Mathematics Subject Classification. 05A19, 11B83, 11C20, 11Y55. Key words and phrases. Bell polynomial of the second kind; closed form; convolved Fibonacci number; determinantal form; Fa`adi Bruno formula; Fibonacci number; Hessenberg determinant; tridiagonal determinant. This paper was typeset using AMS-LATEX. 1 2 M. C. DAGLI˘ AND F. QI in a variety of context. For instance, Ramirez defined in [37] the convolved h(x)- Fibonacci numbers as an extension of classical convolved Fibonacci numbers and offered several combinatorial formulas. In recent years, the second author of this paper and his colleagues have obtained a number of explicit and recursive formulas for some important numbers and poly- nomials via the Fa`adi Bruno formula, some properties of the Bell polynomials of the second kind, and a general derivative formula for a ratio of two differentiable functions. One may consult the papers [15, 16, 17, 21, 24, 28, 29, 30, 36], especially the review and survey article [31], and closely related references therein. A tridiagonal determinant is a determinant whose nonzero elements locate only on the diagonal and slots horizontally or vertically adjacent the diagonal. In other words, a determinant H = jhijjn×n is called a tridiagonal determinant if hij = 0 for all pairs (i; j) such that ji − jj > 1. See [7]. A determinant H = jhijjn×n is called a lower (or an upper, respectively) Hes- senberg determinant if hij = 0 for all pairs (i; j) such that i + 1 < j (or j + 1 < i, respectively). See [13]. In analytic combinatorics and analytic number theory, explaining a sequence of numbers or a sequence of polynomials in terms of a special and simple determinant is an interesting and important direction and sometimes produces significant results. Accordingly, some sequences and polynomials, such as the Bernoulli numbers and polynomials [22, 26], the Euler numbers and polynomials [40], the Delannoy num- bers [15, 20, 21, 23], the Horadam polynomials [27], and the Fibonacci numbers and polynomial [18, 20, 25, 33], have been considered in this framework. Interested readers can also refer to the studies in [9, 25, 34, 35] and closely related references therein. In mathematics, a closed form is a mathematical expression that can be eval- uated in a finite number of operations and may contain constants, variables, four arithmetic operations, and elementary functions, but usually no limit. For system- atic explanation of closed forms, please refer to the article [3] and closely related references therein. In this paper, we will find several closed and determinantal forms for convolved Fibonacci numbers Fn+1(r) by means of the Fa`adi Bruno formula, some properties and special values for the Bell polynomials of the second kind, and the generating function method, while we will give a determinantal formula for convolved Fibonacci numbers Fn+1(r) in terms of tridiagonal determinants by aid of a general derivative formula, which is fundamental but not extensively circulated yet, for the ratio of two differentiable functions. 2. Lemmas In order to prove our main results, we recall several lemmas below. Lemma 1 ([5, pp. 134 and 139]). The Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn;k(x1; x2; : : : ; xn−k+1) for n ≥ k ≥ 0, is defined by 1 `−k+1 `i X n! Y xi B (x ; x ; : : : ; x ) = : n;k 1 2 n−k+1 Q`−k+1 i! 1≤i≤n i=1 `i! i=1 `i2f0g[N Pn i=1i`i=n Pn i=1`i=k CONVOLVED FIBONACCI NUMBERS 3 The Fa`adi Bruno formula can be described in terms of the Bell polynomials of the second kind Bn;k(x1; x2; : : : ; xn−k+1) by n dn X f ◦ h(t) = f (k)(h(t))B h0(t); h00(t); : : : ; h(n−k+1)(t): (2) dtn n;k k=0 Lemma 2 ([5, p. 135]). For n ≥ k ≥ 0, we have 2 n−k+1 k n Bn;k abx1; ab x2; : : : ; ab xn−k+1 = a b Bn;k(x1; x2; : : : ; xn−k+1); (3) where a and b are any complex number. Lemma 3 ([31, Section 1.4]). For n ≥ k ≥ 0; we have (n − k)!n k B (x; 1; 0;:::; 0) = x2k−n: (4) n;k 2n−k k n − k P1 k P1 k Lemma 4 ([22, Lemma 2.4]). Let f(t) = 1 + k=1 akt and g(t) = 1 + k=1 bkt be formal power series such that f(t)g(t) = 1. Then a1 1 0 ··· 0 0 0 a2 a1 1 ··· 0 0 0 a3 a2 a1 ··· 0 0 0 n . .. bn = (−1) . : an−1 an−2 an−3 ··· a1 1 0 an−1 an−2 an−3 ··· a2 a1 1 an an−1 an−2 ··· a3 a2 a1 Lemma 5 ([4, p. 40, Entry 5]). For two differentiable functions p(x) and q(x) 6= 0; we have p q 0 ··· 0 0 p0 q0 q ··· 0 0 p00 q00 2q0 ··· 0 0 k k 1 d p(x) (−1) . = . .. 0 0 : dxk q(x) qk+1 p(k−2) q(k−2) k−2q(k−3) ··· q 0 1 p(k−1) q(k−1) k−1q(k−2) ··· k−1q0 q 1 k−2 (k) (k) k (k−1) k 00 k 0 p q 1 q ··· k−2 q k−1 q In other words, k k d p(x) (−1) = U(k+1)×1(x) V(k+1)×k(x) ; dxk q(x) qk+1(x) (`−1) where the matrix U(k+1)×1(x) = (u`;1(x)) has the elements u`;1(x) = p (x) for 1 ≤ ` ≤ k + 1 and the matrix V(k+1)×k(x) has the entries of the form 8i − 1 < q(i−j)(x); i − j ≥ 0 vij(x) = j − 1 :0; i − j < 0 for 1 ≤ i ≤ k + 1 and 1 ≤ j ≤ k: 4 M. C. DAGLI˘ AND F. QI 3. Main results In this section, we state and prove our main results. Theorem 1. For k ≥ 0 and r 2 R, convolved Fibonacci numbers Fk+1(r) can be computed by k X ` (r)` F (r) = ; k+1 k − ` `! `=0 where n−1 ( Y x(x + 1) ··· (x + n − 1); n ≥ 1 (x)n = (x + k) = 1; n = 0 k=0 denotes the rising factorial, or say, Pochhammer symbol, of x 2 C. Consequently, the Fibonacci numbers Fk+1 for k ≥ 0 can be computed by k X ` F = : k+1 k − ` `=0 Proof. By virtue of (2), (3), and (4), we have k k d 1 X −r−` = h−ri 1 − t − t2 B (−1 − 2t; −2; 0;:::; 0) dtk (1 − t − t2)r ` k;` `=0 k X −r−` 1 = h−ri 1 − t − t2 (−2)`B t + ; 1; 0;:::; 0 ` k;` 2 `=0 k X 1 ! h−ri (−2)`B ; 1; 0;:::; 0 ; t ! 0 ` k;` 2 `=0 k 2`−k X (k − `)!k ` 1 = h−ri (−2)` ` 2k−` ` k − ` 2 `=0 k X 1 ` = k! h−ri (−1)` ` `! k − ` `=0 k X 1 ` = k! (r) ; ` `! k − ` `=0 where n−1 ( Y x(x − 1) ··· (x − n + 1); n ≥ 1 hxin = (x − k) = 1; n = 0 k=0 denotes the falling factorial of x 2 C. So, it follows from (1) that k k 1 d 1 X (r)` ` Fk+1(r) = lim = : (5) k! t!0 dtk (1 − t − t2)r `! k − ` l=0 The proof of Theorem 1 is complete. CONVOLVED FIBONACCI NUMBERS 5 Theorem 2 ([19, Theorem 1]). For k ≥ 0 and r 2 R, convolved Fibonacci numbers Fk+1(r) can be computed by p k ` 1 1 X ` k 5 − 1 Fk+1(r) = p (−1) p (r)`(r)k−`: (6) k! 5 −1 k ` 5 + 1 2 `=0 Consequently, the Fibonacci numbers Fk+1 for k ≥ 0 can be computed by p k ` 1 X ` 5 − 1 Fk+1 = p (−1) p : 5 −1 k 5 + 1 2 `=0 Proof. It is easy to see that 1 1 = p p : 1 − t − t2 5 +1 5 −1 t + 2 2 − t It is not difficult to see that k (`) (k−`) dk 1 X k 1 1 = dtk (t + a)r(b − t)r ` (t + a)r (b − t)r `=0 k k−` X k h−ri` h−rik−`(−1) = ` (t + a)r+` (b − t)r+k−` `=0 k X k (r)` (r)k−` = (−1)` ` (t + a)r+` (b − t)r+k−` `=0 k X k (r)` (r)k−` ! (−1)` ; t ! 0: ` ar+` br+k−` `=0 p p 5 +1 5 −1 Accordingly, considering the first equality in (5), taking a = 2 and b = 2 , and simplifying yield the formula (6).
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