In this study, we define Lichtenberg hybrid quaternions. We give the Binet's formula, the generating functions, exponential generating functions and sum formulas of these quaternions. We find some relations between Jacobsthal hybrid quaternions, Mersenne hybrid quaternions and Lichtenberg hybrid quaternions. Also, Vajda's identity and, as consequences, Catalan's identity, d'Ocagne's identity and Cassini's identity are presented

Morales Gamaliel

Mathematica Moravica

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doi: 10.5937/MatMor2501031MMathematica MoravicaVol. 29, No. 1 (2025), 31–41On the Lichtenberg hybrid quaternionsGamaliel Morales 0000-0003-3164-4434Abstract. In this study, we define Lichtenberg hybrid quaternions.We give the Binet’s formula, the generating functions, exponential gen-erating functions and sum formulas of these quaternions. We findsome relations between Jacobsthal hybrid quaternions, Mersenne hy-brid quaternions and Lichtenberg hybrid quaternions. Also, Vajda’sidentity and, as consequences, Catalan’s identity, d’Ocagne’s identityand Cassini’s identity are presented.1. IntroductionLichtenberg numbers are named after Georg Christoph Lichtenberg, whostudied these numbers in the 17th century. Lichtenberg numbers are denotedby ℓn, defined mathematically by the recurrence ℓn+ℓn−1 = 2n−1 and havethe formℓn =16[(−1)n+1 + 2n+2 − 3].The first few terms of the Lichtenberg sequence are:0, 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, . . . (A000975).The Lichtenberg numbers {ℓn}∞n=0 are defined by the following recurrencerelation(1) ℓn+3 = 2ℓn+2 + ℓn+1 − 2ℓn,with ℓ0 = 0, ℓ1 = 1 and ℓ2 = 2 (see, e.g. [7,12]). Also, the Binet formula forLichtenberg numbers is defined in two different ways, including well-knownsequences of order 2:(2) ℓn =12[2n+2 − (−1)n+23− 1]=12[Jn+2 − 1]and(3) ℓn =13[2n+1 − 1− (−1)n + 12]=13[Mn+1 −(−1)n + 12],2020 Mathematics Subject Classification. Primary: 11B37; Secondary: 11B39, 20G20.Key words and phrases. Hybrid numbers, Jacobsthal numbers, Jacobsthal quaternions,Lichtenberg numbers, Mersenne numbers, Mersenne quaternions.Full paper. Received 14 Aug 2024, accepted 24 Feb 2025, available online 25 Feb 2025.©2025 Mathematica Moravica3132 On the Lichtenberg hybrid quaternionswhere Jn is the n-th Jacobsthal number (see [8]) andMn is the n-th Mersennenumber (see [1]).The relation (3) can be rewritten as ℓn = 13 [Mn+1 − bn], where bn is theusual sequence bn =(−1)n+12 . The Jacobsthal numbers {Jn}∞n=0 are definedby(4) Jn+2 = Jn+1 + 2Jn, J0 = 0, J1 = 1.Also, the Mersenne numbers {Mn}∞n=0 are defined by the following recur-rence relation:(5) Mn+2 = 3Mn+1 − 2Mn, M0 = 0, M1 = 1.Additionally, these exists the following relationships between Mersenne andJacobsthal numbers (see, e.g. [1]):(6) Mn ={3Jn, if n ≡ 0 (mod 2);Jn + 4Jn−1, if n ≡ 1 (mod 2).On the other hand, quaternions are generalizations of complex numbersand its multiplication is not commutative. Quaternions have generally thefollowing form Ψ = ψ0 +ψ1i+ψ2j+ψ3k, where {1, i, j, k} are basis quater-nions and ψm are real numbers for all m = 0, 1, 2, 3. The product rule forunits of quaternions is defined by(7) i2 = j2 = k2 = −1, ijk = −1.The author studied generalized Tribonacci quaternions numbers in theirresearch in [2, 3] and applied it to numerous number sequences. Recently,Özdemir introduced the hybrid numbers and gave some of their properties(see [10]). The set of hybrid numbers isR[i, ε,h] ={ϕ0 + ϕ1i + ϕ2ε+ ϕ3h : ϕr ∈ R, r = 0, 1, 2, 3}.The hybrid product is obtained by distributing the terms to the right, pre-serving the order of multiplication of the units and then writing the valuesof the following substituting each product of units by the equalities i2 = −1,ε2 = 0, h2 = 1 and ih = −hi = ε + i. The table 1 shows us that themultiplication operation with the hybrid numbers is not commutative. Theauthor studied hybrid numbers in their work in [4] and applied it to manynumber sequences of order 2.Table 1. The multiplication table for the basis of R[i, ε,h].× 1 i ε h1 1 i ε hi i −1 1− h ε+ iε ε 1 + h 0 −εh h −(ε+ i) ε 1G. Morales 33Using the Lichtenberg numbers {ℓn}∞n=0, we define Lichtenberg hybridnumbers {hℓn}∞n=0 and Lichtenberg hybrid quaternions {Qhℓn}∞n=0 and givesome of their properties. We prove some theorems about Lichtenberg, Jacob-sthal and Mersenne hybrid quaternions. We find generating functions, expo-nential generating functions, Binet formulas, sum formulas of these numbersand many other relationships between them.2. Lichtenberg hybrid quaternion number {Qhℓn}∞n=0The Jacobsthal hybrid number {Jhn}∞n=0, is defined asJhn = Jn + Jn+1i + Jn+2ε+ Jn+3h, n ≥ 0.Similarly, the Mersenne hybrid number {Mhn}∞n=0, is defined asMhn =Mn +Mn+1i +Mn+2ε+Mn+3h, n ≥ 0.The Binet formulas of the Jacobsthal and Mersenne hybrid numbers as fol-lows:(8) Jhn =13[2n(1 + 2i + 4ε+ 8h)− (−1)n(1− i + ε− h)]and(9) Mhn = 2n(1 + 2i + 4ε+ 8h)− (1 + i + ε+ h).For more details on these two types of hybrid numbers see [13,14].Here, we introduce some properties Lichtenberg hybrid quaternions. Wefind some relations between Lichtenberg hybrid quaternion, Jacobsthal hy-brid quaternion and Mersenne hybrid quaternion.Definition 1. The Lichtenberg hybrid number is defined as follows:(10) hℓn = ℓn + ℓn+1i + ℓn+2ε+ ℓn+3h, n ≥ 0,where {i, ε,h} are hybrid units, and ℓn is the n-th Lichtenberg number.Definition 2. The Lichtenberg hybrid quaternion is defined as follows:(11) Qhℓn = hℓn + hℓn+1i+ hℓn+2j + hℓn+3k, n ≥ 0,where {i, j, k} are quaternion units, and hℓn is the n-th Lichtenberg hybridnumber.The Lichtenberg hybrid quaternion can be written as follows:Qhℓn = hℓn + hℓn+1i+ hℓn+2j + hℓn+3k= ℓn + ℓn+1i + ℓn+2ε+ ℓn+3h+ (ℓn+1 + ℓn+2i + ℓn+3ε+ ℓn+4h) i+ (ℓn+2 + ℓn+3i + ℓn+4ε+ ℓn+5h) j+ (ℓn+3 + ℓn+4i + ℓn+5ε+ ℓn+6h) k= Qℓn +Qℓn+1i +Qℓn+2ε+Qℓn+3h,34 On the Lichtenberg hybrid quaternionswhere Qℓn = ℓn + ℓn+1i + ℓn+2j + ℓn+3k is the classic n-th Lichtenbergquaternion number.Theorem 1. For all n ≥ 0, the Binet’s formula of the Lichtenberg hybridquaternion is given by:(12) Qhℓn =16[(−1)n+1a1b1 + 2n+2a2b2 − 3a3b3],wherea1 = 1− i+ ε− h, b1 = 1− i+ j − k,a2 = 1 + 2i+ 4ε+ 8h, b2 = 1 + 2i+ 4j + 8k,a3 = 1 + i+ ε+ h, b3 = 1 + i+ j + k.Proof. Using Binet formula of Lichtenberg number ℓn and (10), we havehℓn = ℓn + ℓn+1i + ℓn+2ε+ ℓn+3h=16[(−1)n+1 + 2n+2 − 3]+16[(−1)n+2 + 2n+3 − 3]i+16[(−1)n+3 + 2n+4 − 3]ε+16[(−1)n+4 + 2n+5 − 3]h=16[(−1)n+1a1 + 2n+2a2 − 3a3],where a1 = 1− i+ε−h, a2 = 1+2i+4ε+8h and a3 = 1+ i+ε+h. Now,using Binet formula of Lichtenberg hybrid number hℓn and (11), we haveQhℓn = hℓn + hℓn+1i+ hℓn+2j + hℓn+3k=16[(−1)n+1a1 + 2n+2a2 − 3a3]+16[(−1)n+2a1 + 2n+3a2 − 3a3]i+16[(−1)n+3a1 + 2n+4a2 − 3a3]j+16[(−1)n+4a1 + 2n+5a2 − 3a3]k=16[(−1)n+1a1b1 + 2n+2a2b2 − 3a3b3],where b1 = 1 − i + j − k, b2 = 1 + 2i + 4j + 8k and b3 = 1 + i + j + k.Thus, the proof is completed. □Using Theorem 1, we can deduce the following result.G. Morales 35Corollary 1. For the n-th Lichtenberg hybrid quaternion, we have(13) Qhℓn =12[QJhn+2 − a3b3],where QJhn is the n-th Jacobsthal hybrid quaternion.Theorem 2. The following equation is provided for Lichtenberg hybrid quater-nion:(14) Qhℓn +Qhℓn−1 = 2na2b2 − a3b3.Proof. Using (12), we haveQhℓn +Qhℓn−1 =16[(−1)n+1a1b1 + 2n+2a2b2 − 3a3b3]+16[(−1)na1b1 + 2n+1a2b2 − 3a3b3]=16[3 · 2n+1a2b2 − 6a3b3]= 2na2b2 − a3b3.Thus, the proof is completed. □Remark 1. The sequence 2na2b2 − a3b3 in Theorem 2 is recently studiedby Özkan and Uysal in [11]. Furthermore, this sequence is called Mersennehybrid quaternion and is denoted here as QMhn. Other references on thehybrid quaternions are [9, 15].Remark 2. A general sequence of order 2 is introduced by Daǧdevirenand Kürüz (see [5]) considering hybrid and quaternion numbers. Also, theauthors defined the Horadam hybrid quaternions and presented some of theirproperties. In fact, this new quaternion system is a combination of complexquaternions, hyperbolic quaternions and dual quaternions, and it can beviewed as a generalization of these quaternion systems (see [6]).Theorem 3. The recurrence relation of the Lichtenberg hybrid quaternionas follows:(15) Qhℓn+3 = 2Qhℓn+2 +Qhℓn+1 − 2Qhℓn, n ≥ 0.Proof. We will use (1) and (11) for the proof. We haveQhℓn+3 = hℓn+3 + hℓn+4i+ hℓn+5j + hℓn+6k= 2hℓn+2 + hℓn+1 − 2hℓn+(2hℓn+3 + hℓn+2 − 2hℓn+1)i+(2hℓn+4 + hℓn+3 − 2hℓn+2)j+(2hℓn+5 + hℓn+4 − 2hℓn+3)k= 2(hℓn+2 + hℓn+3i+ hℓn+4j + hℓn+5k)+ hℓn+1 + hℓn+2i+ hℓn+3j + hℓn+4k36 On the Lichtenberg hybrid quaternions− 2(hℓn + hℓn+1i+ hℓn+2j + hℓn+3k)= 2Qhℓn+2 +Qhℓn+1 − 2Qhℓn.Thus, the proof is completed. □Theorem 4. The generating function of the Lichtenberg hybrid quaternionis given by:(16)∞∑n=0Qhℓnxn =Qhℓ0 + (Qhℓ1 − 2Qhℓ0)x+ (Qhℓ2 − 2Qhℓ1 −Qhℓ0)x21− 2x− x2 + 2x3.Proof. Suppose that the generating function of the Lichtenberg hybrid quater-nion sequence has the form g(Qhℓn;x) = Qhℓ0 + Qhℓ1x + Qhℓ2x2 + · · · +Qhℓnxn + · · · . Now, multiplying (1) by xn+3 and then taking summationover 0 to ∞, we get0 =∞∑n=0xn+3(Qhℓn+3 − 2Qhℓn+2 −Qhℓn+1 + 2Qhℓn)= g(Qhℓn;x)−Qhℓ0 −Qhℓ1x−Qhℓ2x2− 2[g(Qhℓn;x)−Qhℓ0 +Qhℓ1x]x−[g(Qhℓn;x)−Qhℓ0]x2+ 2[g(Qhℓn;x)]x3g(Qhℓn;x) =Qhℓ0 + (Qhℓ1 − 2Qhℓ0)x+ (Qhℓ2 − 2Qhℓ1 −Qhℓ0)x21− 2x− x2 + 2x3.Thus, the proof is completed. □Theorem 5. The exponential generating function of Lichtenberg hybridquaternion has the form:(17)∞∑n=0Qhℓn(xnn!)=16(−a1b1 + 4a2b2e3x − 3a3b3e2x)e−x.Proof. Using Binet formula of the Lichtenberg hybrid quaternion Qhℓn inTheorem 1, we have∞∑n=0Qhℓn(xnn!)=16∞∑n=0[(−1)n+1a1b1 + 2n+2a2b2 − 3a3b3] xnn!=16[−a1b1∞∑n=0(−x)nn!+ 4a2b2∞∑n=0(2x)nn!− 3a3b3∞∑n=0xnn!]=16[−a1b1e−x + 4a2b2e2x − 3a3b3ex]=16[−a1b1 + 4a2b2e3x − 3a3b3e2x]e−x.G. Morales 37Thus, the proof is completed. □Theorem 6. The sum of the Lichtenberg hybrid quaternion is given by:(18)n∑r=0Qhℓr =12[Qhℓn+2 − (n+ 2)a3b3 − c1b3 − c2] ,where a3, b3 as in Theorem 1, c1 = i + 4ε + 9h, and c2 = (1 + 3i + 5ε +11h)i+ (4 + 8i+ 16ε+ 32h)j + (9 + 19i+ 37ε+ 75h)k.Proof. Using (9), we haven∑r=0ℓn =12[n+2∑r=0Jr − (n+ 2)]=14[Jn+4 − (2n+ 5)]=12[ℓn+2 − (n+ 2)],where Jn is the n-th Jacobsthal number. Then, we can obtain the nextresultn∑r=0hℓr =n∑r=0[ℓr + ℓr+1i + ℓr+2ε+ ℓr+3h]=n∑r=0ℓr +(n+1∑r=0ℓr)i +(n+2∑r=0ℓr − 1)ε+(n+3∑r=0ℓr − 3)h=12[hℓn+2 − (n+ 2)a3 − c1],where c1 = i + 4ε+ 9h and a3 as in Theorem 1.Finally, using (11), we haven∑r=0Qhℓr =n∑r=0[hℓr + hℓr+1i+ hℓr+2j + hℓr+3k]=n∑r=0hℓr +(n+1∑r=0hℓr − hℓ0)i+(n+2∑r=0hℓr − hℓ0 − hℓ1)j+(n+3∑r=0hℓr − hℓ0 − hℓ1 − hℓ2)k=12[Qhℓn+2 − (n+ 2)a3b3 − c1b3 − c2],where c2 = (1+3i+5ε+11h)i+(4+8i+16ε+32h)j+(9+19i+37ε+75h)kand b3 as in Theorem 1. Thus, the proof is completed. □38 On the Lichtenberg hybrid quaternionsTheorem 7. For the n-th Lichtenberg hybrid quaternion, the following equa-tions are provided:(19) Qhℓn+2 = Qhℓn+1 + 2Qhℓn + a3b3and(20) Qhℓn =16{2QMhn+1 − a1b1 − a3b3, if n ≡ 0 (mod 2);2QMhn+1 + a1b1 − a3b3, if n ≡ 1 (mod 2);where QMhn is the n-th Mersenne hybrid quaternion.Proof. Using (12), we will prove the identity in (19). ThenQhℓn+1 + 2Qhℓn + a3b3 =16[(−1)n+2a1b1 + 2n+3a2b2 − 3a3b3]+16[2(−1)n+1a1b1 + 2n+3a2b2 − 6a3b3]+ a3b3=16[(−1)n+3a1b1 + 2n+4a2b2 − 3a3b3]= Qhℓn+2.Similarly, the proof of (20) can be done. □For simplicity of notation, let(21) Hℓn = (−1)n+1a1b1 + 2n+2a2b2.Then, the Binet formula of the Lichtenberg hybrid quaternions is given by(22) Qhℓn =16[Hℓn − 3a3b3],where Hℓ0 = −a1b1 + 4a2b2 and Hℓ1 = a1b1 + 8a2b2.The Vajda’s identity for the sequence {Hℓn}n≥0 and Lichtenberg hybridquaternion sequence {Qhℓn}n≥0 are given in the next theorem.Theorem 8. Let n, u and v be integers such that n ≥ 0, u ≥ 0 and n+v ≥ 0.Then, we have(23)Hℓn+uHℓn+v −HℓnHℓn+u+v= 3(−2)n+2Ju[2va1b1a2b2 − (−1)va2b2a1b1],(24)Qhℓn+uQhℓn+v −QhℓnQhℓn+u+v=112{(−2)n+2Ju[2va1b1a2b2 − (−1)va2b2a1b1]+ a3b3Kℓn(u)−Kℓn+v(u)a3b3},where Kℓn(u) = Hℓn −Hℓn+u.G. Morales 39Proof.(23): Using (21), a1, a2, b1 and b2 as in Theorem 1, we haveHℓn+uHℓn+v −HℓnHℓn+u+v=((−1)n+u+1a1b1 + 2n+u+2a2b2) ((−1)n+v+1a1b1 + 2n+v+2a2b2)−((−1)n+1a1b1 + 2n+2a2b2) ((−1)n+u+v+1a1b1 + 2n+u+v+2a2b2)= 3(−2)n+2Ju[2va1b1a2b2 − (−1)va2b2a1b1],where Ju is the u-th Jacobsthal number. Note that a1b1a2b2 ̸= a2b2a1b1in above equation.(24): By formula (22) and (23), we get36[Qhℓn+uQhℓn+v −QhℓnQhℓn+u+v]=[Hℓn+u − 3a3b3][Hℓn+v − 3a3b3]−[Hℓn − 3a3b3][Hℓn+u+v − 3a3b3]= Hℓn+uHℓn+v −HℓnHℓn+u+v + 3[a3b3Kℓn(u)−Kℓn+v(u)a3b3]= 3(−2)n+2Ju[2va1b1a2b2 − (−1)va2b2a1b1]+ 3[a3b3Kℓn(u)−Kℓn+v(u)a3b3],where Kℓn(u) = Hℓn −Hℓn+u. □It is easily seen that for special values of u and v by Theorem 8, we getnew identities for Lichtenberg hybrid quaternions:• Catalan’s identity: v = −u and n ≥ u.• Cassini’s identity: u = 1, v = −1 and n ≥ 1.• d’Ocagne’s identity: u = 1, v = m− n and m ≥ n.Corollary 2. Catalan identity for Lichtenberg hybrid quaternions. Let n ≥0, u ≥ 0 be integers such that n ≥ u. ThenQhℓn+uQhℓn−u −Qhℓ2n=112{(−2)n+2Ju[2−ua1b1a2b2 − (−1)−ua2b2a1b1]+ a3b3Kℓn(u)−Kℓn−u(u)a3b3}.Corollary 3. Cassini identity for Lichtenberg hybrid quaternions. Let n ≥ 1be an integer. ThenQhℓn+1Qhℓn−1 −Qhℓ2n=112{(−2)n+1[a1b1a2b2 + 2a2b2a1b1]+ a3b3Kℓn(1)−Kℓn−1(1)a3b3}.40 On the Lichtenberg hybrid quaternionsCorollary 4. d’Ocagne identity for Lichtenberg hybrid quaternions. Letn ≥ 0, m ≥ 0 be integers such that m ≥ n. ThenQhℓn+1Qhℓm −QhℓnQhℓm+1=112{(−2)n+2[2m−na1b1a2b2 − (−1)m−na2b2a1b1]+ a3b3Kℓn(1)−Kℓm(1)a3b3}.3. Conclusions and remarksIn this paper, we discussed the Lichtenberg hybrid quaternions and theirproperties. We obtained the Binet’s formula, the generating function, expo-nential generating function and sum formula of these numbers. Further, wefound some relations between Lichtenberg hybrid quaternion, Jacobsthal hy-brid quaternion and Mersenne hybrid quaternion. In the future, the study ofquadratic properties in this type of numbers could be encouraged, includingthe study of hybrid numbers with Lichtenberg hybrid number coefficients.AcknowledgementThe author is thankful to the referee for valuable suggestions leading toimproving the quality of the paper.References[1] P. Catarino, H. Campos and P. Vasco, On the Mersenne sequence, Annales Mathe-maticae et Informaticae, 46 (2016), 37–53.[2] G. Cerda-Morales, Identities for third order Jacobsthal quaternions, Advances in Ap-plied Clifford Algebras, 27 (2) (2017), 1043–1053.[3] G. Cerda-Morales, On a generalization of Tribonacci quaternions, MediterraneanJournal of Mathematics, 14 (2017), Article ID: 239, 1–12.[4] G. Cerda-Morales, Investigation of generalized Fibonacci hybrid numbers and theirproperties, Applied Mathematics E-Notes, 21 (2021), 110–118.[5] A. Daǧdeviren and F. Kürüz, On the Horadam hybrid quaternions, Advanced Studies:Euro-Tbilisi Mathematical Journal, 10 (2022), 293–306.[6] A. Daǧdeviren, A generalization of complex, dual and hyperbolic quaternions: hybridquaternions, Filomat, 37 (25) (2023), 8441–8454.[7] A.M. Hinz, The Lichtenberg sequence, Fibonacci Quarterly, 55 (2) (2017), 2–12.[8] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quarterly, 43 (1)(1996), 40–54.[9] M.C.S. Mangueira, F.R.V. Alves and P.M.M.C. Catarino, Hybrid quaternions ofLeonardo, Trends in Computational and Applied Mathematics, 23 (1) (2022),51–62.G. Morales 41[10] M. Özdemir, Introduction to hybrid numbers, Advances in Applied Clifford Algebras,28 (2018), 1–32.[11] E. Özkan and M. Uysal, On the Mersenne and Mersenne-Lucas hybrid quaternions,Bulletin of the Transilvania University of Brasov, Series III: Mathematics and Com-puter Science, 4 (1) (2024), 139–148.[12] P.K. Stockmeyer, An exploration of sequence A000975, Fibonacci Quarterly, 55 (5)(2017), 174–185.[13] A. Szynal-Liana and I. Włoch, On Jacobsthal and Jacobsthal-Lucas hybrid numbers,Annales Mathematicae Silesianae, 33 (2019), 276–283.[14] D. Taşci and E. Sevgi, Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas hybrid numbers, Chaos, Solitons & Fractals, 146 (2021), Article ID: 110862,1–7.[15] M. Uysal and E. Özkan, Padovan hybrid quaternions and some properties, Journalof Science and Arts, 22 (1) (2022), 121–132.Gamaliel MoralesInstituto de MatemáticasPontificia Universidad Católica de ValparaísoBlanco Viel 596, ValparaísoChileE-mail address: gamaliel.cerda.m@mail.pucv.cl

On the Lichtenberg hybrid quaternions

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On the Lichtenberg hybrid quaternions

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