Cobalancing hybrid numbers (Q6595843)

\textit{A. Behera} and \textit{G. K. Panda} [Fibonacci Q. 37, No. 2, 98--105 (1999; Zbl 0962.11014)] defined that a positive integer \(n\) is called a balancing number if the Diophantine equation\N\[\N1+2+\cdots +(n-1)=(n+1)+(n+2)+\cdots +(n+r)\N\]\Nholds for some positive integer \(r\) which is called balancer. \textit{G. K. Panda} and \textit{P. K. Ray} [Int. J. Math. Math. Sci. 2005, No. 8, 1189--1200 (2005; Zbl 1085.11017)] defined that a positive integer \(n\) is called a cobalancing number if the Diophantine equation\N\[\N1+2+\cdots +n=(n+1)+(n+2)+\cdots +(n+r)\N\]\Nholds for some positive integer \(r\) which is called cobalancer.\N\NLet \(B_{n}\) denote the \(n^{\text{th}}\) balancing number and let \(b_{n}\) denote the \(n^{\text{th}}\) cobalancing number. Then from above equalities, \(B_{n}\) is a balancing number if and only if \( 8B_{n}^{2}+1\) is a perfect square and \(b_{n}\) is a cobalancing number if and only if \(8b_{n}^{2}+8b_{n}+1\) is a perfect square. Thus \(C_{n}=\sqrt{8B_{n}^{2}+1}\) and \(c_{n}=\sqrt{8b_{n}^{2}+8b_{n}+1}\) are integers which are called the Lucas-balancing number and Lucas-cobalancing number, respectively.\N\NThe balancing and Lucas-balancing hybrid numbers were introduced in [\textit{D. Bród} et al., Indian J. Math. 63, No. 1, 143--157 (2021; Zbl 1471.11053)] as follows. For a nonnegative integer \(n\), the \(n^{\text{th}}\) balancing hybrid number \(BH_{n}\) is defined as\N\[\NBH_{n}=B_{n}+B_{n+1}i+B_{n+2}\varepsilon +B_{n+3}h,\N\]\Nwhere \(B_{n}\) denotes the \(n^{\text{th}}\) balancing number. Similarly, the \(n^{\text{th}}\) Lucas-balancing hybrid number \(CH_{n}\) is defined as\N\[\NCH_{n}=C_{n}+C_{n+1}i+C_{n+2}\varepsilon +C_{n+3}h,\N\]\Nwhere \(C_{n}\) denotes the \(n^{\text{th}}\) Lucas-balancing number and \(i^{2}=-1,\) \(\varepsilon ^{2}=0,\) \(h^{2}=1,\) \(ih=-hi=\varepsilon +i\).\N\NIn that paper, the authors, considered the cobalancing hybrid numbers and Lucas-cobalancing hybrid numbers defined as\N\[\NbH_{n}=b_{n}+b_{n+1}i+b_{n+2}\varepsilon +b_{n+3}h\N\]\Nand\N\[\NcH_{n}=c_{n}+c_{n+1}i+c_{n+2}\varepsilon +c_{n+3}h\N\]\Nrespectively, where \(b_{n}\) denotes the \(n^{\text{th}}\) cobalancing number and \(c_{n}\) denotes the \(n^{\text{th}}\) Lucas-cobalancing number.\N\NThey proved that for \(n\geq 2\),\N\[\NbH_{n}=6bH_{n-1}-bH_{n-2}+2(1+i+\varepsilon +h),\N\]\Nwhere \(bH_{0}=2\varepsilon +14h,bH_{1}=2i+14\varepsilon +84h\) and\N\[\NcH_{n}=6cH_{n-1}-cH_{n-2},\N\]\Nwhere \(cH_{0}=-1+i+7\varepsilon +41h\) and \(cH_{1}=1+7i+41\varepsilon +239h\). Further, they deduced Binet formulas to be\N\[\NbH_{n}=\frac{\alpha ^{n-\frac{1}{2}}}{\alpha -\beta }\widetilde{\alpha }- \frac{\beta ^{n-\frac{1}{2}}}{\alpha -\beta }\widetilde{\beta }-\frac{1}{2} \widetilde{1}\N\]\Nand\N\[\NcH_{n}=\frac{\alpha ^{n-\frac{1}{2}}\widetilde{\alpha }+\beta ^{n-\frac{1}{2}}\widetilde{\beta }}{2},\N\]\Nwhere \(\alpha =3+2\sqrt{2},\) \(\beta =3-2\sqrt{2}\), \(\widetilde{\alpha }=1+\alpha i+\alpha ^{2}\varepsilon +\alpha ^{3}k\), \(\widetilde{\beta } =1+\beta i+\beta ^{2}\varepsilon +\beta ^{3}k\) and \(\widetilde{1} =1+i+\varepsilon h\).\N\NAlso, they derived some similar fundamental identities on general bilinear index-reduction formulas which imply the Catalan, Cassini, Vajda, d'Ocagne and Halton identities and the generating functions for these hybrid numbers.