This article considers a generalized Fibonacci sequence {Vn} with general initial conditions, V0 = a, V1 = b, and a versatile recurrence relation Vn = pVn-1 + qVn-2, where n ≥ 2 and a, b, p and q are any non-zero real numbers. The generating function and Binet formula for this generalized sequence are derived. This generalization encompasses various well-known sequences, including their generating functions and Binet formulas as special cases. Numerous new properties of these sequences are studied and investigated

Verma K.L.

Mathematica Moravica

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doi: 10.5937/MatMor2501113VMathematica MoravicaVol. 29, No. 1 (2025), 113–124The mathematics of generalized Fibonaccisequences: Binet’s formula and identitiesK.L. Verma∗ 0000-0002-6486-8736Abstract. This article considers a generalized Fibonacci sequence {Vn}with general initial conditions, V0 = a, V1 = b, and a versatile recur-rence relation Vn = pVn−1+qVn−2, where n ≥ 2 and a, b, p and q are anynon-zero real numbers. The generating function and Binet formula forthis generalized sequence are derived. This generalization encompassesvarious well-known sequences, including their generating functions andBinet formulas as special cases. Numerous new properties of these se-quences are studied and investigated.1. IntroductionSeveral generalizations of the Classical Fibonacci sequence are availablein the literature. Some authors ([2,11,13–15,19,25]) have altered the initialconditions, while others ([4, 16–18, 20–22, 24, 26, 27]) have generalized therecurrence relation. Author [8] presented an analytic proof of the classicalFibonacci recursive formula. Fibonacci and Lucas Numbers are studied in [9]and [10] using different approaches. Some generalized formulas of Fibonacciare also obtained and studied in [7].This article considers a generalization of the Fibonacci sequence by mod-ifying both the initial conditions and the recurrence relation. By applyingappropriate restrictions to the parameters, this generalization encompassesthe Classical Fibonacci, Lucas, Pell, Pell-Lucas, modified Pell, Goksal Bil-gici, Jacobstal, and Jacobstal-Lucas sequences. These generalizations ex-hibit numerous fascinating properties and have applications in various fieldsof mathematical sciences, including quasi-crystals, algebra, graph theory,and computer algorithms ([1, 3, 5, 6, 14,21,23]).Definition 1 (Fibonacci’s Generalization). We define the generalization ofthe Fibonacci sequence {Vn}∞n=0 by the following recurrence relations:(1) Vn = pVn−1 + qVn−22020 Mathematics Subject Classification. 11B37; 11B39; 11B83.Key words and phrases. Generalized Fibonacci sequence, Recurrence relation, Binet for-mula, Identities.Full paper. Received 4 Jan 2025, accepted 28 May 2025, available online 20 June 2025.©2025 Mathematica Moravica113114 Mathematics of generalized Fibonacci sequenceswith the initial conditions,V0 = a, V1 = b, a, b, p and q are any non–zero realnumbers.The expression for {Vn} in (1) is true [15] for every integer n ≥ 2.Definition 2 (Fibonacci’s Recursion Generalization representation). Equiv-alently, expression for {Vn} in (1) can be expressed as:(2)Vn =[n−12 ]∑i=0(n− i− 1i)pn−2i−1qi b+[n−22 ]∑i=0(n− i− 2i)pn−2i−2qi a, n > 0.,a, b, p and q are any non–zero real numbers.Remark 1. If n ≥ 2 is any integer, and p = 1, q = 1 and V0 = F0 = 0,V1 = F1 = 1 in (2), thenVn =[n−12 ]∑i=0(n− i− 1i)pn−2i−1qi , n > 0.Evidently, for (p, q) = (1, 1) and (a, b) = (1, 1), (p, q) = (1, 1) and (a, b) =(2, 1) , (p, q) = (2, 1) and (a, b) = (2, 1) , (p, q) = (2, 1) and (a, b) = (2, 2) ,(p, q) = (1, 2) and (a, b) = (0, 1) , (p, q) = (1, 2) and (a, b) = (2, 1) and(p, q) =(2s, t2 − s)and (a, b) = (0, 1) , (p, q) =(2s, t2 − s)and (a, b) =(2, 2s) , where s and t are any non – zero real numbers, the sequence {Vn}defined in (1) and (2) becomes the Classical Fibonacci, Lucas, Pell, ModifiedPell, Pell-Lucas Jacobsthal, Jacobsthal-Lucas and Goksal Bilgici sequences,respectively.The well-known Classical Fibonacci sequence is defined by the recur-rence relation and initial conditions isDefinition 3 (Classical Fibonacci’s Recursion). Fn = Fn−1 + Fn−2, n ≥ 2,F0 = 0, F1 = 1.Some of the well–known generalizations of the classical Fibonacci se-quences are:• The Lucas sequence is defined as: Ln = Ln−1 + Ln−2, n ≥ 2,L0 = 2, L1 = 1.• The Pell sequence is defined as: Pn = 2Pn−1+Pn−2, n ≥ 2, P0 = 2,P1 = 1. qn = 2qn−1 + qn−2, n ≥ 2, q0 = 1, 11 = 1.• The Pell-Lucas sequence is defined is defined as: Qn = 2Qn−1 +Qn−2, n ≥ 2, Q0 = 2, Q1 = 2.K.L. Verma 115• The Goksal Bilgici sequences are defined by the recurrence rela-tion, fn = 2aqn−1 + (b2 − a)qn−2, n ≥ 2, q0 = q, q1 = 1.and ln = 2aln−1 + (b2 − a)ln−2, n ≥ 2, l0 = 2, q1 = 2a.• The Jacobsthal sequences are defined as: Jn = Jn−1 + 2Jn−2,n ≥ 2, J0 = 0, J1 = 1.• The Jacobsthal-Lucas sequences are defined as: Ln = Ln−1 +2Jn−2, n ≥ 2, L0 = 2, L1 = 1.Theorem 1. If n ≥ 2 is any integer and {Vn} defined in (2), thenn∑i=0(nr)Vr =1(α− β)pn(p− q + α2)n(V1 − βV0)−(p− q + β2)n(V1 − αV0) .Proof. Using the definition of {Vn} in (2), we haven∑i=0(nr)Vr =1(α− β)[(1 + α)n (V1 − βV0)− (1 + β)n (V1 − αV0)].Now using1 + α =(p− q + α2)/p, 1 + β =(p− q + β2)/p,we haven∑i=0(nr)Vr =1(α− β)pn(p− q + α2)n(V1 − βV0)−(p− q + β2)n(V1 − αV0) . □Corollary 1. If p = 1, q = 1 and V0 = F0 = 0, V1 = F1 = 1 is substitutedin the above expression, thenn∑i=0(nr)Fr =1(α− β)(α2)n − (β2)n = α2n − β2nα− β= F2n,which is in agreement with (Lucas) (page 157, Theorem 12.5 of ThomasKoshy [15]).Theorem 2. Ifn ≥ 2, is any integer and {Vn} defined in (2), thenn∑i=0(nr)(−1)rVr =1(α− β)[(1− α)n (V1 − βV0)− (1− β)n (V1 − αV0)] .Proof. Using the definition of {Vn} from (2), we haven∑i=0(nr)(−1)rVr =1(α− β)(1− α)n (V1 − βV0)−1(α− β)(1− β)n (V1 − αV0)=1(α− β)[(1− α)n (V1 − βV0)− (1− β)n (V1 − αV0)],where α and β are the roots of the equation x2 − px− q = 0. □116 Mathematics of generalized Fibonacci sequencesCorollary 2. If p = 1, q = 1 and V0 = F0 = 0, V1 = F1 = 1 are substitutedin the above expression, thenn∑i=0(nr)(−1)rFr =1(α− β)[(−β)n − (−α)n]= (−1)n−1(αn − βnα− β)= (−1)n−1Fn,where α and β are the roots of the equation x2 − x− 1 = 0.Theorem 3 (Generalized Generating Functions). The generalized generat-ing function of the sequence defined in (1) is∞∑n=0Vnxn =a+ (b− pa)x(1− px− qx2).Proof. Let V (x) be the generating function of (1),(3) V (x) =∞∑n=0Vnxn,then(4) V (x)− pxV (x)− qx2V (x) =∞∑n=0Vnxn − px∞∑n=0Vnxn − qx2∞∑n=0Vnxn,On simplification, we have(1− px− qx2)V (x) = V0 + (V1 − pV0)xHence□(5)∞∑n=0Vnxn =a+ (b− pa)x(1− px− qx2).Theorem 4 (Generalized Binet’s formula). The generalized Binet’s formulafor the sequence {Vn} in (2) isVn = an∑k=0αn−kβk + (b− a (α+ β))n∑k=0αn−k−1βkwhere α, β = p±√p2+4q2 are the roots of the equation x2 − px− q = 0.Proof. Consider partial fraction decomposition of the right-hand side of thegenerating function of the sequence defined in (5)a+ (b− pa)x(1− px− qx2)≡ (αa+ b− pa)(α− β) (1− αx)− (βa+ b− pa)(α− β) (1− βx).On simplification we have=a(α− β)[α(1− αx)− β(1− βx)]+(b− pa)(α− β)[1(1− αx)− 1(1− βx)]K.L. Verma 117we have(6)∞∑n=0Vnxn =[a(αn+1 − βn+1α− β)+ (b− pa)(αn − βnα− β)]xn.Thus, the generalized form of the Binet formula for the generalized Fibonaccisequence {Vn}∞n=0 is(7) Vn = a(αn+1 − βn+1α− β)+ (b− pa)(αn − βnα− β),or equivalently expression (3) can be expressed as(8) Vn = an∑k=0αn−kβk+(b− a (α+ β))n∑k=0αn−k−1βk,where α and β are the roots of the equation x2 − px− q = 0. □2. Special Cases of generating functions and Binet formulaCase 1: Fibonacci sequencesSubstituting p = 1, q = 1 and a = F0 = 0, b = F1 = 1 in (3) and (4),subsequently, the generating function and Binet formula simplifies to:Fn = Fn−1 + Fn−2 (Vn = pVn−1 + pVn−2), n ≥ 2 is(9)∞∑n=0Fnxn =F0 + (F1 − pF0)x(1− px− qx2)=1(1− x− x2),(10) Vn = 0(αn+1 − βn+1α− β)+ (1− 0)(αn − βnα− β)Therefore, (9) and (10) represent the generating function and Binet formulafor the well-known classical Fibonacci sequences.Case 2: Lucas sequenceSubstituting p = 1, q = 1 and a = l0 = 2, b = l1 = 1 in (3), then thegenerating function in (3), simplifies to:ln = 2ln−1 + ln−2, n ≥ 2 is(11)∞∑n=0lnxn =2− x(1− x− x2),(12) ln = αn + βn,where α, β = 1±√52 . Thus, (11) and (12) represent the generating functionand Binet formula for the well-known Lucas sequence. This is in agreementwith the generating function for Lucas sequence.Case 3: Pell sequenceSubstituting p = 2, q = 1 and a = P0 = 0, b = P1 = 1 in (3), then thegenerating function in (3), simplifies to:118 Mathematics of generalized Fibonacci sequencesPn = 2Pn−1 + Pn−2, n ≥ 2 is(13)∞∑n=0Pnxn =1(1− 2x− x2).(14) Pn =αn − βnα− β, α, β = 1±√2.Thus, (13) and (14) represent the generating function and Binet formula forthe Pell sequence. This is in agreement with the generating function andBinet formula for Pell sequence.Case 4:Modified Pell sequenceSubstituting p = 2, q = 1 and a = f0 = 1, b = f1 = 1 in (3), then thegenerating function in (3), simplifies to:fn = 2fn−1 + fn−2, n ≥ 2 is(15)∞∑n=0fnxn =1− x(1− 2x− x2).(16) qn = αn + βn, α, β = 1±√2.Thus, (15) and (16) represent the generating function and Binet formulafor the Modified Pell sequence. This is in agreement with the generatingfunction and Binet formula for Modified Pell sequence.Case 5: Pell-Lucas sequence Substituting p = 2, q = 1 and a = Q0 = 2,b = Q1 = 2 in (3), then the generating function in (3) simplifies to:Qn = 2Qn−1 +Qn−2, n ≥ 2 is(17)∞∑n=0fnxn =2− 2x(1− 2x− x2).(18) ⇒ Qn = 2[(αn+1 − βn+1α− β)−(αn − βnα− β)]= 2qn, α, β = 1±√2.Thus, (17) and (18) represent the generating function and Binet formula forthe Pell-Lucas sequence. This is in agreement with the generating functionand Binet formula for the Pell-Lucas sequence.Case 6: Goksal Bilgici sequencesSubstituting p = 2a, q = b− a2 and V0 = f0 = 0, V1 = f1 = 1and p = 2a, q = b − a2 and V0 = l0 = 2, V1 = l1 = 2a in (3), then thecorresponding generating functions for the Goksal Bilgici sequences are:fn = 2afn−1 + (b− a2)fn−2, n ≥ 2andln = 2aln−1 + (b− a2)ln−2, n ≥ 2K.L. Verma 119are(19)∞∑n=0fnxn =x(1− 2ax− (b− a2)x2)and(20)∞∑n=0lnxn =2− 2ax(1− 2ax− (b− a2)x2).Thus, (19) and (20) are in agreement with the generating function for thewell-known Goksal Bilgici sequences(21) fn =αn − βnα− βand(22) ln = αn + βn.Thus, (21) and (22) are in agreement with the Binet’s formul for the GoksalBilgici sequences.Case 7: Jacobsthal Sequences Substituting p = 1, q = 2 and a = J0 = 0,b = J1 = 1 in (3), then the corresponding generating functions for theJacobsthal sequence is(23)∞∑n=0Jnxn =x(1− x− 2x2),(24)Ln = 2(αn+1 − βn+1α− β)− αn − βnα− β,Ln = αn + βn (∵ 2α− 1 = 3, 2β − 1 = −3, α− β = 3) .Here α and β are roots of the equation x2 − x − 2 = 0, this formula is thesame as is in [12]. Thus, (23) and (24) represent the generating function andBinet formula for the Jacobsthal Sequence. This is in agreement with thegenerating function and Binet formula for the Jacobsthal Sequence.Case 8: Jacobsthal-Lucas Sequences Substituting p = 1, q = 2 and a = J0 =2, b = J1 = 1 in (3), then the corresponding generating functions for theJacobsthal-Lucas sequence are(25)∞∑n=0Jnxn =2− x(1− x− 2x2),(26) Ln = αn + βn (∵ 2α− 1 = 3, 2β − 1 = −3, α− β = 3) .This is in agreement with the generating function and Binet formula for theJacobsthal Sequence. Here α and β are roots of the equation x2−x−2 = 0,this formula is the same as is in [12]. Thus, (25) and (26) represent thegenerating function and Binet formula for the Jacobsthal Sequence. □120 Mathematics of generalized Fibonacci sequencesTheorem 5. If n ≥ 2 is any integer andVn =[n−12 ]∑i=0(n− i− 1i)pn−2i−1qi b+[n−22 ]∑i=0(n− i− 2i)pn−2i−2qi a,where a, b, p and q are any non–zero real numbers andKn =(Vn+2 Vn+1Vn+1 Vn),thendetK = (−1)n(qn−2b(b− ap) + qn−1a2).Corollary 3. On Substituting p = 1, q = 1 and V0 = F0 = 0, V1 = F1 = 1,the above results reduce todetK = (−1)n.Theorem 12.4. (Lucas Formula, 1876). If a = 0, b = 1, p = q = 1, thenV (n) reduces to Lucas formula, 1876 (see page 175].Theorem 6. If n ≥ 2 is any integer and {Vn} defined in (2), thenn∑i=0(nr)(−1)rVr+m=1(α− β)[(1− α)nβm (V1 − βV0)− (1− β)nαm (V1 − αV0)]Proof. We haven∑i=0(nr)(−1)rVr+m=1(α− β)(1− α)n (V1 − βV0)βm −1(α− β)(1− β)n (V1 − αV0)αm=1(α− β)[(1− α)nαm (V1 − βV0)− (1− β)nβm (V1 − αV0)],where α and β are the roots of the equation x2 − px− q = 0. □K.L. Verma 121Corollary 4. On substituting p = 1, q = 1 and a = F0 = 0, b = F1 = 1,the above results are reduced todetK = (−1)n,n∑i=0(nr)(−1)rFr+m = (−1)m+1(αn−m − βn−mα− β)= (−1)m+1Fn−m,where α and β are the roots of the equation x2 − x− 1 = 0.Theorem 7. If n ≥ 2 is any integer and {Vn} defined in (2), then∞∑n=2(1Vn−1− 1Vn+1)=1V1+1V2=1b+1pb+ aqProof. We have∞∑n=2(1Vn−1− 1Vn+1)=(1V1− 1V3)+(1V2− 1V4)+(1V3− 1V5)+ · · ·=1V1+1V2=1b+1pb+ aq. □Corollary 5. On substituting p = 1, q = 1 and a = F0 = 0, b = F1 = 1,the above results are reduced to∞∑n=2(1Fn−1− 1Fn+1)=1F1+1F2=11+11= 2.Theorem 8. If n ≥ 1 is any integer then using the Binet’s formula in (4),we haveVn − αVn−1 = βn−1 (b− αa) ,Vn − βVn−1 = αn−1 (b− βa) ,where α and β are the roots of the equation x2 − px− q = 0.Proof. Using the Binet’s formula (4), for n ≥ 1 we haveVn − αVn−1 = a(αn+1 − βn+1α− β)+ (b− (α+ β) a)(αn − βnα− β)− α(a(αn − βnα− β)+ (b− (α+ β) a)(αn−1 − βn−1α− β)).On simplification we obtainVn − αVn−1 = βn−1 (b− αa) .SimilarlyVn − βVn−1 = αn−1 (b− βa) ,where α and β are the roots of the equation x2 − px− q = 0 . □122 Mathematics of generalized Fibonacci sequencesCorollary 6. On substituting p = 1, q = 1 and a = F0 = 0, b = F1 = 1, inthe above expression we obtain the corresponding expression for n-th classicalFibonacci number [5].Fn − αFn−1 = βn−1,Fn − βFn−1 = αn−1,where α and β are the roots of the equation x2 − x− 1 = 0.Theorem 9 (Linearization). If n ≥ 1 is any integer, then using the Binet’sformula in (4) we haveβn =1V1(βVn + qVn−1 − qa) ,αn =1V1(αVn + qVn−1 − qa) ,where α and β are the roots of the equation x2 − px− q = 0.Proof. Using the expressions obtained in the above theorem,Vn − αVn−1 = βn−1 (V1 − αa) ,Vn − βVn−1 = αn−1 (V1 − βa) .We haveβn =1V1(βVn + qVn−1 − qa) ,αn =1V1(αVn + qVn−1 − qa) ,where α and β are the roots of the equation x2 − px− q = 0. □Corollary 7. On substituting p = 1, q = 1 and a = F0 = 0, b = F1 = 1, inthe above expression we obtain the corresponding expression for n-th classicalFibonacci number [5]Fn − αFn−1 = βn−1,Fn − βFn−1 = αn−1,where α and β are the roots of the equation x2 − x− 1 = 0 .3. Discussion and conclusionIn mathematics, the most fascinating sequence of number is the ClassicalFibonacci sequence. For mathematicians, it offers various opportunities fornew inferences. In this paper, an advanced generalization of the Fibonaccisequence is introduced, where unlike other generalizations, its parametersa, b, p, and q for initial conditions and recurrence relations can be anyreal numbers. Considering this generalization of the Fibonacci sequence,we present the generalized sequence’s term formula, generating function,K.L. Verma 123Binet’s formula and some well-known identities in their generalized form.Any reader can employ this generalization for any existing or new identities.References[1] J. Atkins and R. Geist, Fibonacci numbers and computer algorithms, The CollegeMathematics Journal, 18 (1987), 328-337.[2] G. Bilgici, New generalizations of Fibonacci and Lucas sequences, Applied Mathe-matical Sciences, 8 (29) (2014), 1429-1437.[3] P. 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Shin, The Binet formula and representationsof k-generalized Fibonacci numbers, The Fibonacci Quarterly, 39 (2) (2001), 158-164.[19] J.C. Pond, Generalized Fibonacci Summations, The Fibonacci Quarterly, 6 (1968),97-108.124 Mathematics of generalized Fibonacci sequences[20] S.P. Pethe and C.N. Phadte, A generalization of the Fibonacci sequence, Applicationsof Fibonacci Numbers, 5 (1992), 465-472.[21] G. Sburlati, Generalized Fibonacci sequences and linear congruence, The FibonacciQuarterly, 40 (2002), 446-452.[22] I. Stojmenovic, Recursive Algorithms in Computer Science Courses: Fibonacci Num-bers and Binomial Coefficients, IEEE Transactions on Education, 43 (2000), 273-276.[23] J. de Souza, E M.F. Curado and M.A. Rego-Monteiro, Generalized Heisenberg Alge-bras and Fibonacci Series, Journal of Physics A: Mathematical and General, 39 (33)(2006), ArticleID: 10415.[24] J.E. Walton and A.F. Horadam, Some further identities for the generalized Fibonaccisequence Hn, The Fibonacci Quarterly, 12 (1974), 272-280.[25] M. Waddill and L. Sacks, Another generalized Fibonacci sequence, The FibonacciQuarterly, 5 (3) (1967), 209-222.[26] O. Yayenie, A note on generalized Fibonacci sequences, Applied Mathematics andComputation, 217 (12) (2011), 5603-5611.[27] L.J. Zai and L.J. Sheng, Some properties of the generalization of the Fibonacci se-quence, The Fibonacci Quarterly, 25 (2) (1987), 111-117.K.L. VermaDepartment of MathematicCareer Point UniversityHamirpur, 176041IndiaE-mail address: klverma@netscape.netklverma@cpuh.edu.in

The mathematics of generalized Fibonacci sequences: Binet's formula and identities

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The mathematics of generalized Fibonacci sequences: Binet's formula and identities

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