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Rayaguru, S. G.
- Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers
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Authors
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela, IN
1 Department of Mathematics, National Institute of Technology, Rourkela, IN
Source
The Journal of the Indian Mathematical Society, Vol 86, No 1-2 (2019), Pagination: 137-160Abstract
In this article, we obtain the closed form expressions for different types of summation formulas involving certain powers of balancing and Lucas-balancing numbers using the telescoping summation formula.Keywords
Balancing Numbers, Lucas-balancing Numbers, Telescoping Summation Formula.
References
- Adegoke, K., Factored closed-form expressions for the sums of cubes of Fibonacci and Lucas numbers, https://arxiv.org/pdf/1706.08469.pdf.
- Adegoke, K., Sums of fourth powers of Fibonacci and Lucas numbers, https://arxiv.org/pdf/1706.00407.pdf.
- Behera, A., Panda, G.K., On the square ischolar_mains of triangular numbers, Fib. Quart., 37(2)(1999), 98–105.
- Clary, S., Hemenway, P.D., On sums of cubes of Fibonacci numbers, in Applications of Fibonacci Numbers, Kluwer Academic Publishers, Dordrecht, The Netherlands, 5 (1993), 123–136.
- Davala, R.K., Panda, G.K., On Sum and Ratio Formulas for Balancing Numbers, Journal of the Ind. Math. Soc., 82(1–2) (2015).
- Melham, R.S., Alternating sums of fourth powers of Fibonacci and Lucas numbers, Fib. Quart., 38(3) (2000), 254–259.
- Panda, G.K., Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (2009), 185–189.
- Shift Balancing Numbers
Abstract Views :261 |
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Authors
Affiliations
1 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
2 Department of Mathematics, Gayatri Vidya Parishad College of Engineering (A), Visakhapatnam - 530048, IN
1 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
2 Department of Mathematics, Gayatri Vidya Parishad College of Engineering (A), Visakhapatnam - 530048, IN
Source
The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 131–136Abstract
For each positive integer k, the Diophantine equation (k+1)+(k+2)+···+(n−1) = (n+1)+(n+2)+···+(n+r) is studiedKeywords
Lancing Numbers, Lucas-Balancing Numbers, Gap Balancing Numbers, t−balancing Numbers.References
- A. Behera and G. K. Panda, On the square ischolar_mains of triangular numbers, Fibonacci Quart., 37(2) (1999), 98-105.
- K. K. Dash and R. S. Ota, t−Balancing Numbers, Int. J. Contemp. Math. Sciences, 7 (2012), 1999- 2012.
- R. K. Davala and G. K. Panda, On sum and ratio formulas for balancing numbers, J. Ind. Math. Soc., 82(1-2) (2015), 23-32.
- R. P. Finkelstein, The House Problem, Amer. Math. Monthly, 72 (1965), 1082-1088.
- T. Kovács, K. Liptai and P. Olajos, On (a,b)-balancing numbers, Publ. Math. Debrecen, 77(3) (2010).
- K. Liptai, Fibonacci Balancing Numbers, Fibonacci Quart., 42(4) (2004), 330-340.
- K. Liptai, Lucas balancing numbers, Commun. Math., 14(1) (2006), 43-47.
- R. A. Mollin, Fundamental number theory with applications, Boca Raton, CRC press, London (2004).
- G. K. Panda and R. K. Davala, Perfect balancing numbers, Fibonacci Quart., 53(3) (2015), 261-264.
- G. K. Panda and P. K. Ray, Cobalancing numbers and cobalancers, Internat. J. Math. Math. Sci., 8 (2005), 1189-1200.
- G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, (2009), 185-189.
- G. K. Panda and A. K. Panda, Almost balancing numbers, J. Ind. Math. Soc., 82(3-4) (2015), 147-156.
- G. K. Panda and S. S. Rout, Periodicity of balancing numbers, Acta. Math. Hungar., 143(2) (2014), 274-286.
- S. S. Rout and G. K. Panda, k-gap balancing numbers, Mathematika, 70(1) (2015), 109-121.
- S. S. Rout, R. K. Davala and G. K. Panda, Stability of balancing sequence modulo p, Unif. Distrib. Theory, 10(2) (2015), 77-91.
- Brousseau’s Reciprocal Sums Involving Balancing and Lucas-Balancing Numbers
Abstract Views :103 |
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Authors
Affiliations
1 Department of Center for Data Science, Siksha ’O’ Anusandhan Deemed to be University, Bhubaneswar - 751 030, IN
2 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
1 Department of Center for Data Science, Siksha ’O’ Anusandhan Deemed to be University, Bhubaneswar - 751 030, IN
2 Department of Mathematics, National Institute of Technology, Rourkela - 769 008, IN
Source
The Journal of the Indian Mathematical Society, Vol 89, No 1-2 (2022), Pagination: 145–166Abstract
In this paper, we derive the closed form expressions for the finite and infinite sums with summands having products of balancing and Lucas-balancing numbers in the denominator. We present some generalized Brousseau’s sums for balancing and Lucas-balancing numbers.
Keywords
Balancing numbers, Lucas-balancing numbers, gap balancing numbers, t-balancing numbers.References
- K. Adegoke, Generalizations of the Reciprocal Fibonacci-Lucas Sums of Brousseau, J. Integer Seq., Vol. 21 (2018), Article 18.1.6.
- A. Behera and G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37(2) (1999), 98–105.
- Bro. A. Brousseau, Summation of Infinite Fibonacci series, Fib. Quart., 7 (1969), 143–168.
- Bro. A. Brousseau, Fibonacci-Lucas Infinite Series-Research Topic, Fib. Quart., 7 (1969), 211–217.
- R. K. Davala and G. K. Panda, On Sum and Ratio Formulas for Balancing Numbers, J. Indian Math. Soc., 82(1-2) (2015), 23–32.
- R. Frontczak, New results on reciprocal series related to Fibonacci and Lucas numbers with subscripts in arithmetic progression, Int. J. Contemporary Math. Sci., 11 (2016), 509–516.
- G. K. Panda, Some fascinating properties of balancing numbers, In Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194 (2009), 185–189.
- S. G. Rayaguru and G. K. Panda, Some infinite product identities involving balancing and Lucas-balancing numbers, Alabama J. Math., 42 (2018).
- S. G. Rayaguru and G. K. Panda, Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers, J. Indian Math. Society, 86(1-2) (2019), 1–24.
- S. G. Rayaguru and G. K. Panda, Sum Formulas Involving Powers of Balancing and Lucas-balancing Numbers-II, Notes on Number Theory and Discrete Mathematics, 25(3) (2019), 102–110.