Ratio Mathematica Volume 40, 2021, pp. 163-177 Generalized double Fibonomial numbers Shah Mansi S.* Shah Devbhadra V.† Abstract From the beginning of 20th century, generalization of binomial coef- ficient has been deliberated broadly. One of the most famous gener- alized binomial coefficients are Fibonomial coefficients, obtained by substituting Fibonacci numbers in place of natural numbers in the bi- nomial coefficients. In this paper, we further generalize the concept of Fibonomial coefficient and called it Generalized double Fibono- mial number and obtain interesting properties of it. We also discuss its special case, double Fibonomial number along with the situation in which they give integer values. Other properties of it have also been discussed along with its upper and lower bounds. Keywords: Fibonacci numbers, Lucas numbers, Fibonomial num- bers, Binomial coefficient, Double factorial. 2020 AMS subject classifications: 11B39,05A10,11B65. 1 *Veer Narmad South Gujarat University, Surat, India; mansi.shah 88@yahoo.co.in. †Veer Narmad South Gujarat University, Surat, India; drdvshah@yahoo.com. 1Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. doi: 10.23755/rm.v40i1.625. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published under the CC-BY licence agreement. 163 M. Shah, D. Shah 1 Introduction In combinatorics, the factorial of a positive integer n, denoted by n!, is defined by n! = n×n−1×···×2×1;n ≥ 1 and 0! = 1. Whereas the double factorial of a positive integer n, usually denoted by n!! is defined as n!! =   n×n−2×···×3×1; n is odd n×n−2×···×4×2; n is even 1; n = 0 Note that n!! is not the same as the iterated factorial (n!)!, which grows much faster. We do not know precisely when, where, or by whom, the double factorial notation was devised. It was used by Meserve [6] in 1948, and it is not mentioned by Cajori in his very detailed work in history of mathematical notations during 1928 – 1929 [2]. Thus, we summaries that the notation was introduced at some times during the period 1928 – 1948. In this definition of n!!, if we replace the natural numbers by the terms of the generalized Fibonacci numbers wn defined by the recurrence relation wn = pwn−1 + qwn−2, for n ≥ 2; w0 = a and w1 = b, where a,b,p and q are any integers, then what we get will be called Generalized double Fibonorial n!!w and is defined as n!!w ≡   wn ×wn−2 ×···×w3 ×w1 n > 0 is odd wn ×wn−2 ×···×w4 ×w2 n > 0 is even 1 n = 0 (1) Here note that when we substitute p = q = b = 1 and a = 0 in the definition of wn, we get regular Fibonacci numbers Fn. The definition of n!!w helps to express the generalized double Fibonorial in terms of regular generalized Fibonorial as shown in the following lemma. Lemma 1.1. n!!w = n!w(n−1)!!w = (n+1)!w (n+1)!!w ;n ≥ 1. In 1964 Fontene [3] generalized the notion of binomial coefficients and intro- duce the new concept of Fibonomial coefficients. In the definition of binomial coefficients ( m k ) , he replaced the natural numbers by the terms of an arbitrary sequence {An} of real or complex numbers. Since then there has been an accel- erated interest in the study of Fibonomial coefficients. When the sequence {An} is considered as the sequence {Fn} of Fibonacci numbers, the Fibonomial coeffi- cients [ m k ] F , for 1 ≥ k ≥ m, is defined as [ m k ] F = m!F k!F (m−k)!F . The elaborated study on the generalized Fibonomial coefficients can be found in literature. (See [5]) 164 Generalized double Fibonomial numbers This quantity will always be an integer, which can be shown by an induction argument by replacing Fm in the numerator with FkFm−k+1+Fk−1Fm−k, resulting in [ m k ] F = Fm−k+1 [ m−1 k −1 ] F + Fk−1 [ m−1 k ] F (2) We use the concept of generalized double Fibonorial to further generalize the concept of generalized Fibonomial coefficients. We define the generalized double Fibonomial numbers [[ m k ]] w as [[ m k ]] w = m!!w k!!w (m−k)!!w (3) It is easy to observe that[[ m 0 ]] w = 1, [[ m 2 ]] w = wmand [[ m k ]] w = [[ m m−k ]] w (4) 2 Generalized double Fibonomial numbers: 2.1 Some properties of generalized double Fibonomial num- bers: The following results are now easy consequences from this definition: Lemma 2.1. For any positive integers k,m and n, 1. (Iterative rule) [[ n k ]] w [[ k m ]] w = [[ n m ]] w [[ n−m k −m ]] w . 2. wm−k [[ m k ]] w = wm [[ m−2 k ]] w . 3. wk [[ m k ]] w = wm−k+2 [[ m k −2 ]] w . 4. wk [[ m k ]] w = wm [[ m−2 k −2 ]] w . Lemma 2.2. (m−1)!!w [[ m k ]] w will always give an integer value. 165 M. Shah, D. Shah This result is an easy derived from the definition of generalized Fibonorial and generalized double Fibonomial numbers. The basic recurrence relations for the generalized double Fibonomial numbers is as follows: Lemma 2.3. [[ m k ]] w − [[ m−2 k ]] w = [[ m−2 k −2 ]] w { wm−wm−k wk } . By changing k to m−k and using (4), we get Lemma 2.4. [[ m k ]] w − [[ m−2 k −2 ]] w = [[ m−2 k ]] w { wm−wk wm−k } . The following result can be easily obtained when we apply the sum on both sides with respect to the upper index such that m and k have the same parity. Lemma 2.5. [[ m k ]] w = ∑m j=k { wj−wj−k wk }[[ j −2 k −2 ]] w ; where the sum is taken over integers starting from k with spacing of 2 up to m. 2.2 Star of David theorem: In 1972, Gould gave a result related to one interesting arithmetic property of binomial coefficients which was named as the Star of David theorem, which was stated as “The greatest common divisors of the binomial coefficients forming each of the two triangles in the Star of David shape in Pascal’s triangle are equal: gcd {( n−1 k−1 ) , ( n k+1 ) , ( n+1 k )} . The two sets of three numbers, which the Star of David theorem says, have equal greatest common divisors and equal products. Interestingly, Gould’s result can be imitated for generalized double Fibonomial numbers too as shown in the following result. Theorem 2.1. [[ m−a k − b ]] w [[ m k + b ]] w [[ m + b k ]] w = [[ m−a k ]] w [[ m + b k + a ]] w [[ m k − b ]] w ; where a,b are positive integers. Proof. Using the definition of generalized double Fibonomial numbers, the left side of the result becomes[[ m−a k − b ]] w [[ m k + b ]] w [[ m + b k ]] w = (m−a)!!w (k−b)!!w(m−k−a+b)!!w × (m)!!w (k+a)!!w(m−k−a)!!w × (m+b)!!w (k)!!w(m−k+b)!!w 166 Generalized double Fibonomial numbers = (m−a)!!w (k)!!w(m−k−a)!!w × (m+b)!!w (k+a)!!w(m−k−a+b)!!w × (m)!!w (k−b)!!w(m−k+b)!!w = [[ m−a k ]] w [[ m + b k + a ]] w [[ m k − b ]] w , as required.2 Corolary 2.1. If a = b = 1, we get the product of six generalized double Fibono- mial numbers, which are equally spaced around [[ m k ]] w . 2.3 Generalized Double Multinomial Numbers: Let m = k1 +k2 +· · ·+kr then we can define generalized double multinomial number as [[ m k1,k2, · · · ,kr ]] w = m!!w k1!!wk2!!w···kr!!w Following result expresses generalized double multinomial numbers as the multi- plication of generalized double Fibonomial numbers. Lemma 2.6. Generalized double multinomial numbers can be expressed as the multiplication of generalized double Fibonomial numbers. Proof. In the definition of generalized double multinomial numbers, consider r = 2, then we have [[ m k1,k2 ]] w = [[ m k1 ]] w ; where k1 + k2 = m. For r = 3 and m = k1 + k2 + k3, [[ m k1,k2,k3 ]] w = [[ m k1 ]] w [[ m−k1 k2 ]] w . Let us now consider r = n and m = k1 + k2 + · · ·+ kn. Thus[[ m k1,k2, · · · ,kr ]] w = m!!w k1!!wk2!!w···kn!!w = m!!w k1!!wk2!!w···kn−2!!w × 1 kn−1!!wkn!!w = m!!w k1!!wk2!!w···kn−2!!w×(m−k1−k2−···−kn−2)!!w × (m−k1−k2−···−kn−2)!!w kn−1!!w(m−k1−k2−···−kn−2−kn−1)!!w = [[ m k1 ]] w [[ m−k1 k2 ]] w · · · [[ m−k1 −k2 −···−kn−2 kn−1 ]] w . Hence, by the principle of Mathematical induction, we get the required result.2 It is obvious that all the above results related to generalized double Fibonorials and generalized double Fibonomial numbers are also true for double Fibonomials n!!F and double Fibonomial coefficients [[ m k ]] F . But there are some additional results related to them, which are discussed in the following article. 167 M. Shah, D. Shah 3 Double Fibonomial numbers: 3.1 Definition and some properties of double Fibonomial num- bers: Using the definitions (1) and (3), Double Fibonorials and double Fibonomial numbers can be respectively expressed as n!!F ≡   Fn ×Fn−2 ×···×F3 ×F1 n > 0 is odd Fn ×Fn−2 ×···×F4 ×F2 n > 0 is even 1 n = 0 and [[ m k ]] F = m!!F k!!F (m−k)!!F . The following table shows first few terms of double Fibonorials for some initial values of n. n 0 1 2 3 4 5 6 7 8 9 10 n!!F 1 1 1 2 3 10 24 130 504 4420 27720 Table 1: Double Fibonorial numbers Also by (4), double FIbonomial numbers have the symmetry property. Thus Table 2 shows the first few terms of double Fibonomial numbers of one side only. 1 1 1 1 1 2 1 3/2 3 1 10/3 5 1 24/10 8 6 1 65/12 13 65/3 1 252/65 21 126/5 56 Table 2: Double Fibonomial numbers We further show how Double Fibonorial and Double Fibonomial numbers are connected with the sequence {Ln} of Lucas numbers. This sequence is famously known as the twin sequence of Fibonacci sequence, which can be obtained by 168 Generalized double Fibonomial numbers substituting p = q = b = 1 and a = 2 in the definition of wn. That is Ln = Ln−1 + Ln−2;L0 = 2 and L1 = 1. It is easy to observe that F2n = FnLn. If we define n!L = Ln×Ln−1×···×L2×L1, then the following lemma follows easily. Lemma 3.1. n!!F = k!F ×k!L, for even positive integer n = 2k. If we consider [ m k ] L = m!L k!L(m−k)!L ,then the following is an easy consequence of lemma 3.1. Lemma 3.2. [[ 2m 2k ]] F = [ m k ] F × [ m k ] L From the Table 2 it is clear that double Fibonomial numbers are not always an integer. Obviously, for any integer m, [[ m 0 ]] F = [[ m m ]] F = 1, will always have an integer value. Also [[ m 2 ]] F = [[ m m−2 ]] F = Fm will be integers. These two will serve as the trivial cases. Following theorem speaks about when double Fibonomial numbers attain integer values. Theorem 3.1. The nontrivial double Fibonomial numbers [[ m k ]] F are integers only when either m and k both are even integers together or [[ m k ]] F = [[ 6 3 ]] F . Proof. We prove the result in four cases depending on the parity of m and k. Case 1: When m and k both are even integers, we have[[ m k ]] F = [[ 2n 2l ]] F = (2n)!!F (2l)!!F (2n−2l)!!F = F2n×F2n−2×···×F2n−2l+2 F2l×···×F4×F2 Note that number of elements in numerator and denominator are same. Also, they are Fibonacci numbers with even subscripts, such that in the denominator we have first l even subscripted Fibonacci numbers. Since these numbers always divide multiplication of any l consecutive even subscripted Fibonacci numbers, it follows that [[ m k ]] F will always be an integer. Case 2: When m and k both are odd integers, we have In this case, we have[[ m k ]] F = [[ 2n + 1 2l + 1 ]] F = (2n+1)!!F (2l+1)!!F (2n−2l)!!F = F2n+1×F2n−1×···×F2l+3 F2n−2l×···×F4×F2 169 M. Shah, D. Shah In the numerator, every Fibonacci number is with odd subscript. Consequently, none of them will be divisible by F4 = 3. Thus [[ m k ]] F will not be an integer in this case. Case 3: When m is odd integer and k is even integer, we have In this case, we have[[ m k ]] F = [[ 2n + 1 2l ]] F = (2n+1)!!F (2l)!!F (2n−2l+1)!!F = F2n+1×F2n−1×···×F2n−2l+3 F2l×···×F4×F2 Here again in the numerator, every Fibonacci number is with odd subscript, so none of them will be divisible by F4 = 3. And therefore, in this case [[ m k ]] F will not be an integer. Case 4: When m is even integer and k is odd integer, we have[[ m k ]] F = [[ 2n 2l + 1 ]] F = (2n)!!F (2l+1)!!F (2n−2l−1)!!F = F2n×F2n−2×···×F4×F2 (F2l+1×···×F3×F1)(F2n−2l−1×···×F3×F1) Here number of terms in the numerator and denominator are same. Also, the Fibonacci numbers in the numerator are with only even subscripts and in the de- nominator with only odd subscripts. But, for any Fibonacci number Fn, there exists a prime p such that if p | Fn, then p will only divide Fmn; for every m ≥ 1. Since [[ m k ]] F = [[ m m−k ]] F , for convenience we take k > m−k, that is, 2k > m. Then there will not be the same Fibonacci numbers in the numerator and denominator. Also, there will not be any multiple subscripts of k in the numerator. Thus, there will exist a prime p in the denominator such that p | Fk which will not divide any of the Fibonacci number in the numerator. Likewise, when k = m−k, then except for k = 3, there will be a prime p such that p | Fk as well as p | Fm, but it will appear in the denominator only once where as in the numerator twice. Thus in this case, except for [[ 6 3 ]] F = 6, [[ m k ]] F will not be an integer.2 In the following theorem we obtain the recurrence relation for the double Fi- bonomial numbers. Theorem 3.2. [[ m k ]] F = Fk−1 [[ m−2 k ]] F + Fm−k+1 [[ m−2 k −2 ]] F . Proof. From [4], we observe that the Fibonomial coefficients [ m k ] F has the recurrence relation 170 Generalized double Fibonomial numbers [ m k ] F = Fk−1 [ m−1 k ] F + Fm−k+1 [ m−1 k −1 ] F Now, using this relation and lemma 3.2, we get[[ m k ]] F = m!F (m−1)!!F × (k−1)!!F k!F × (m−k−1)!!F (m−k)!F = [ m k ] F × (k−1)!!F×(m−k−1)!!F (m−1)!!F . = { Fk−1 [ m−1 k ] F + Fm−k+1 [ m−1 k −1 ] F } × (k−1)!!F×(m−k−1)!!F (m−1)!!F . = { Fk−1(m−1)!F (m−1)!!F × (k−1)!!F k!F × (n−k−1)!!F (n−k−1)!F } + { Fn−k+1(m−1)!F (m−1)!!F × (k−1)!!F (k−1)!F × (n−k−1)!!F (n−k−1)!F } = Fk−1(n−2)!!F k!!F×(n−k−2)!!F + Fn−k+1(n−2)!!F (k−2)!!F×(n−k)!!F[[ m k ]] F = Fk−1 [[ m−2 k ]] F + Fm−k+1 [[ m−2 k −2 ]] F , as required. 2 Lemma 3.3. [[ m k ]] F = ∑|m−k2 | j=1 F j−1 k−1Fm−k+1−2(j−1) [[ m−2j k −2 ]] F + F |m−k2 | k−1 A; where A =   1;when m and k both are even or both are odd integers[[ k + 1 k ]] F ;otherwise Proof. From above theorem, we have[[ m k ]] F = Fm−k+1 [[ m−2 k −2 ]] F + Fk−1 [[ m−2 k ]] F = Fm−k+1 [[ m−2 k −2 ]] F + Fk−1 { Fm−k−1 [[ m−4 k −2 ]] F + Fk−1 [[ m−4 k ]] F } = Fm−k+1 [[ m−2 k −2 ]] F + Fk−1Fm−k−1 [[ m−4 k −2 ]] F . + F2k−1 { Fm−k−3 [[ m−6 k −2 ]] F + Fk−1 [[ m−6 k ]] F } Continuing this process, we get [[ m k ]] F =   ∑|m−k2 | j=1 F j−1 k−1Fm−k+1−2(j−1) [[ m−2j k −2 ]] F + F |m−k2 | k−1 [[ k k ]] F ; when n and k both are even or odd∑|m−k2 | j=1 F j−1 k−1Fm−k+1−2(j−1) [[ m−2j k −2 ]] F + F |m−k2 | k−1 [[ k + 1 k ]] F ; otherwise ,as required. 2 To illustrate the result, we consider m = 9 and k = 5. Then[[ m k ]] F = ∑|m−k2 | j=1 F j−1 k−1Fm−k+1−2(j−1) [[ m−2j k −2 ]] F + F |m−k2 | k−1 A = ∑2 j=1 F j−1 4 F5−2(j−1) [[ 9−2j 3 ]] F + F24 [[ 5 5 ]] F 171 M. Shah, D. Shah = F5 [[ 7 3 ]] F + F4F3 [[ 5 3 ]] F + F24 = ( 5× 65 3 ) + (3×2×5) + (32) = 442 3 = [[ 9 5 ]] F , as expected. The following result is an easy consequence from the definition of double Fibonomial numbers and the basic identity FmLn +FnLm = 2Fm+n relating both Fibonacci numbers and Lucas numbers. Lemma 3.4. [[ m k ]] F = 1 2 ( Lk [[ m−2 k ]] F + Lm−k [[ m−2 k −2 ]] F ) . Proof. Since 2Fm = FkLm−k + Fm−kLk , we have 2 [[ m k ]] F Fm =[[ m k ]] F FkLm−k + [[ m k ]] F Fm−kLk = [[ m−2 k −2 ]] F FmLm−k + [[ m−2 k ]] F FmLk. Thus 2 [[ m k ]] F = Lk [[ m−2 k ]] F + Lm−k [[ m−2 k −2 ]] F , as required. 2 Using lemma 3.4 and applying the same logic of lemma 3.3, the following result can be proved easily. Lemma 3.5. [[ m k ]] F = ∑bm−k2 c j=1 L j−1 k Lm−k−2(j−1) 2j [[ m−2j k −2 ]] F + ( Lk 2 )bm−k2 cA; where A =   1;when m and k both are even or odd integers[[ k + 1 k ]] F ;otherwise To illustrate the result, we consider m = 10 and k = 3. Then [[ m k ]] F = ∑bm−k2 c j=1 L j−1 k Lm−k−2(j−1) 2j [[ m−2j k −2 ]] F + ( Lk 2 )bm−k2 cA = ∑3 j=1 L j−1 3 L7−2(j−1) 2j [[ 10−2j 3 ]] F + ( L3 2 )3 [[4 3 ]] F = L7 2 [[ 8 1 ]] F + L3L5 22 [[ 6 1 ]] F + L23L3 23 [[ 4 1 ]] F + L33 23 [[ 4 3 ]] F = ( 29 2 × 252 65 ) + ( 4×11 22 × 24 10 ) + ( 43 23 )( 3 2 + 3 2 ) = 1386 13 = [[ 10 3 ]] F , as expected. In the following section we find the bounds of these numbers. 172 Generalized double Fibonomial numbers 3.2 Bounds of double Fibonomial numbers: The Binet formula for the Fibonacci number is given by Fn = αn−βn α−β ; where α = 1+ √ 5 2 and β = 1− √ 5 2 . The following theorem gives us the bounds of double Fibonomial numbers in terms of α. Theorem 3.3. For χ(n) = { 0;when n is even 1;when n is odd , α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] F ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . Proof. It is well-known that αn−2 ≤ Fn ≤ αn−1; for all n ≥ 1. Then it is easy to observe that Fm−2t F2t+2 ≤ αm−4t−1 (5) and Fm−2t F2t+2 ≥ αm−4t−3 (6) Here we consider the four cases depending on the parity of m and k. When both m and k are even, using the definition of double Fibonomial numbers and (5), we have[[ m k ]] F = m!!F k!!F×(m−k)!!F = Fm×Fm−2×···×Fm−k+2 F2×F4×···×Fk ≤ αm−1 ×αm−5 ×···×α(m−2k+3) = α k(m−k+1) 2 Thus [[ m k ]] F ≤ α k(m−k+1) 2 . Again using (6) in the definition of double Fibonomial number, we get[[ m k ]] F ≥ αm−3 ×αm−7 ×···×αm−2k+1 = α k(m−k−1) 2 . This shows that [[ m k ]] F ≥ α k(m−k−1) 2 . Thus when m and n both are even, we have α k(m−k−1) 2 ≤ [[ m k ]] F ≤ α k(m−k+1) 2 . Considering χ(n) = { 0;when n is even 1;when n is odd , this result can be written as α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] F ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . 173 M. Shah, D. Shah The required result can be proved using the similar technique for all the remaining cases. To illustrate it, we consider m = 9 and k = 4. Then [[ m k ]] F = 442 3 . Also, α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 = α k(m−k−1) 2 = α8 = 46.97 and α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 = α k(m−k+1) 2 = α12 = 321, which shows that α (k−χ(k))(m−k−χ(m(m−k−1)−1)) 2 ≤ [[ m k ]] F ≤ α (k+χ(k))(m−k+χ(m(m−k−1)−1)) 2 . 4 Double Fibonomial numbers and Fibonacci num- bers: By [1], it is known that a primitive divisor of a Fibonacci number Fn is any prime integer p such that p | Fn but p - Fm; where m < n. Also, primitive divisor theorem says that for n ≥ 13, every Fn has a primitive divisor. We use this result to prove many interesting relations between generalized double Fibonomial numbers and Fibonacci numbers. 4.1 Double Fibonomial number as a power of Fibonacci num- ber: In literature, there are many results involving Fibonomial numbers and Fi- bonacci numbers. From (4), it is clear that [[ m k ]] F = Fm for k = 2. Thus, the Diophantine equation [[ m k ]] F = Fn will always have a trivial solution (m,k,n) = (m,2,m). Following result claims that there is no other solution for the consid- ered Diophantine equation. Lemma 4.1. The Diophantine equation [[ m k ]] F = Fn has no solution for k > 2. Proof. We know that except for [[ 6 3 ]] F = 6, which is not a Fibonacci number, and trivial cases, [[ m k ]] F is an integer only when both m and k are even integers. Thus, [[ m k ]] F = Fn implies Fm ×Fm−2 ×···×Fm−k+2 Fk ×Fk−2 ×···×F2 = Fn (7) 174 Generalized double Fibonomial numbers If we consider n ≥ 13 and n > m, then by the primitive divisor theorem, there exists a prime p such that p | Fn but p - Fm. That is, (7) has no solution possible. Similarly, for m ≥ 13 and m > n, primitive divisor theorem implies that (7) has no solution. Thus, we can narrow down the range of m and n as max(m,n). A quick look at the Table 2 reveals that for k > 2, the Diophantine equation [[ m k ]] F = Fn has no solution. 2 The following result can be proved through the similar arguments. Theorem 4.1. For any positive integer t, the Diophantine equation [[ m k ]] F = F tn has no solution for k > 2. Though the double Fibonomial numbers do not possess the value of a Fi- bonacci number except for the trivial cases, they do stand in the neighborhood of Fibonacci number. We present this fact in the following final result. Theorem 4.2. The only solutions of the Diophantine equation [[ m k ]] F ±1 = Fn are (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) ,(6,3,5) ,(8,4,10) for ′+′ case and (3,1,4) ,(3,2,4) for ′−′ case. Poof. From the Table 2, it is easy to observe that the Diophantine equation[[ m k ]] F ± 1 = Fn has solution (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) for ′+′ case and (m,k,n) = (3,1,4) ,(3,2,4) for the ′−′ case for m ≤ 5. Now for m > 5, when m is an odd integer, double Fibonomial number will not be an integer. And when m is an even integer such that k is an odd integer, [[ 6 3 ]] F = 6 is the only possibility integer value of double Fibonomial. Thus (m,k,n) = (6,3,5) will be a solution of the given Diophantine equation for ′+′ case. Now, we can narrow down our possible solution to the even integers for both m and k. Since FaLb = Fa+b + (−1) b Fa−b, the different factorizations for Fn±1 depending on the class of nmodulo4 can be written as: F4l + 1 = F2l−1L2l+1 F4l −1 = F2l+1L2l−1 F4l+1 + 1 = F2l+1L2l F4l+1 −1 = F2lL2l+1 F4l+2 + 1 = F2l+2L2l F4l+2 −1 = F2lL2l+2 F4l+3 + 1 = F2l+1L2l+2 F4l+3 −1 = F2l+2L2l+1 175 M. Shah, D. Shah Therefore, the considered Diophantine equation, which can also be written as[[ m k ]] F = Fn ∓1, can be factorized for the ′+′ case as[[ m k ]] F = F2l+1L2l−1 [[ m k ]] F = F2lL2l+1 [[ m k ]] F = F2lL2l+2[[ m k ]] F = F2l+2L2l+1; and for the ′−′ case as[[ m k ]] F = F2l−1L2l+1 [[ m k ]] F = F2l+1L2l [[ m k ]] F = F2l+2L2l[[ m k ]] F = F2l+1L2l+2; It is obvious that all these eight cases can be handled in the similar man- ner. Thus, we shall only focus on the proof of the first case. Now, [[ m k ]] F = F2l+1L2l−1 implies Fm×Fm−2×···×Fm−k+2 Fk×Fk−2×···×F2 = F2l+1L2l−1. Thus, we have Fm ×Fm−2 ×···×Fm−k+2 = F2l+1 ×L2l−1 ×Fk ×Fk−2 ×···×F2 Since F2n = FnLn, we write L2l−1 = F4l−2 F2l−1 . Thus Fm ×Fm−2 ×···×Fm−k+2 ×F2l−1 = F2l+1 ×F4l−2 ×Fk ×Fk−2 ×···×F2. Since l = ⌊ n 4 ⌋ > 2, we have4l − 2 > 2l + 1. Therefore, from primitive divisor theorem, we can write m = 4l−2. Thus, Fm−2 ×···×Fm−k+2 ×F2l−1 = F2l+1 ×Fk ×Fk−2 ×···×F2 (8) If we assume that m ≥ max{14,k + 1}, we have m − 2 ≥ 12. So, again by primitive divisor theorem, we get m − 2 = max{2l + 1,k}. But m − 2 = 4l − 4 > 2l + 1, which implies m− 2 = k and from (8), we get F2l−1 = F2l+1, which is not possible. Thus, we only need to consider the range 4 ≤ k ≤ 10 and k + 2 ≤ m ≤ 12. Again, from Table 2, we can easily claim that the only solution of the Diophan- tine equation [[ m k ]] F ±1 = Fn are (m,k,n) = (3,1,2) ,(3,2,2) ,(4,2,3) , (6,3,5) ,(8,4,10) for ′+′ case and (3,1,4) ,(3,2,4) for ′−′ case. 2 5 Conclusion: In this paper, we have defined double Fibonorial numbers and double Fibono- mial numbers. We have proved many properties for these numbers including re- cursive equations in terms of Fibonacci numbers and Lucas numbers. We have 176 Generalized double Fibonomial numbers extended the star of David theorem for double Fibonomial numbers and also dis- cussed various Diophantine equations related to double Fibonomial numbers and Fibonacci numbers. 6 Acknowledgement: The authors are thankful to the Department of Science and Technology for providing financial support under WOS – A fellowship. References [1] BILU, Y., HANROT, G. and VOUTIER, P. M. 2011. Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. [2] CAJORI F. 1993. 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