Prodinger, Helmut Summing a family of generalized Pell numbers. (English) Zbl 1471.11077 Ann. Math. Sil. 35, No. 1, 105-112 (2021). Summary: A new family of generalized Pell numbers was recently introduced and studied by D. Bród [Ann. Math. Sil. 33, 66–76 (2019; Zbl 1470.11022)]. These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power \(P^\ell_n\) is expressed as a linear combination of \(P_{mn}\). The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter \(R=2^r\), the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times. MSC: 11B39 Fibonacci and Lucas numbers and polynomials and generalizations 05A15 Exact enumeration problems, generating functions Keywords:Pell numbers; Binet formula; generating functions Citations:Zbl 1470.11022 Software:OEIS PDFBibTeX XMLCite \textit{H. Prodinger}, Ann. Math. Sil. 35, No. 1, 105--112 (2021; Zbl 1471.11077) Full Text: DOI arXiv References: [1] M. Bicknell, A primer on the Pell sequence and related sequences, Fibonacci Quart. 13 (1975), no. 4, 345-349. · Zbl 0319.10013 [2] D. Bród, On a new one parameter generalization of Pell numbers, Ann. Math. Sil. 33 (2019), 66-76. · Zbl 1470.11022 [3] J.M. Mahon and A.F. Horadam, Ordinary generating functions for Pell polynomials, Fibonacci Quart. 25 (1987), no. 1, 45-56. · Zbl 0609.10006 [4] The online encyclopedia of integer sequences. http://oeis.org. · Zbl 1274.11001 [5] T. Komatsu, Higher-order identities for balancing numbers, Notes Number Theory Discrete Math. 26 (2020), no. 2, 71-84. [6] T. Mansour and M. Shattuck, Restricted partitions and q-Pell numbers, Cent. Eur. J. Math. 9 (2011), no. 2, 346-355. · Zbl 1238.05021 [7] H. Prodinger, On a sum of Melham and its variants, Fibonacci Quart. 46/47 (2008/2009), no. 3, 207-215. · Zbl 1220.11023 [8] H. Prodinger, Sums of powers over equally spaced Fibonacci numbers, Integers 20 (2020), paper A37, 6 pp. · Zbl 1472.11070 [9] H. Prodinger, How to sum powers of balancing numbers efficiently, arXiv preprint 2020. Avaliable at arXiv:2008.03916. [10] L. Trojnar-Spelina and I. Włoch, On generalized Pell and Pell-Lucas numbers, Iran. J. Sci. Technol. Trans. A Sci. 43 (2019), no. 6, 2871-2877. · Zbl 1463.11053 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.