By means of the recursive construction method, we examine a class of multiple sums (with a free variable "$ x $") for circular products of binomial coefficients. They are first expressed as coefficients of bivariate rational functions. Then the rational generating functions are determined by deftly employing Knuth's bracket calculus, resultants of polynomials, and Hadamard products of rational functions.
Citation: Marta Na Chen, Wenchang Chu. Generating functions for circular sums of binomial products[J]. AIMS Mathematics, 2025, 10(12): 28182-28206. doi: 10.3934/math.20251239
By means of the recursive construction method, we examine a class of multiple sums (with a free variable "$ x $") for circular products of binomial coefficients. They are first expressed as coefficients of bivariate rational functions. Then the rational generating functions are determined by deftly employing Knuth's bracket calculus, resultants of polynomials, and Hadamard products of rational functions.
| [1] |
J. P. Allouche, M. Mendé France, Hadamard grade of power series, J. Number Theory, 131 (2011), 2013–2022. https://doi.org/10.1016/j.jnt.2011.04.011 doi: 10.1016/j.jnt.2011.04.011
|
| [2] | A. T. Benjamin, J. A. Rouse, In: F. T. Howard, Recounting binomial Fibonacci identities, Applications of Fibonacci numbers, Springer, 2004, 25–28. https://doi.org/10.1007/978-0-306-48517-6_4 |
| [3] |
A. Bostan, P. Flajolet, B. Salvy, É. Schost, Fast computation of special resultants, J. Symbolic Comput., 41 (2006), 1–29. https://doi.org/10.1016/j.jsc.2005.07.001 doi: 10.1016/j.jsc.2005.07.001
|
| [4] |
L. Carlitz, The characteristic polynomial of a certain matrix of binomial coefficients, Fibonacci Quart., 3 (1965), 81–89. https://doi.org/10.1080/00150517.1965.12431433 doi: 10.1080/00150517.1965.12431433
|
| [5] | M. N. Chen, W. Chu, Triple symmetric sums of circular binomial products, Mathematics, 12 (2024), 2303. https://doi.org/10.3390/math12152303 |
| [6] | W. Chu, Circular sums of binomial coefficients, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 115 (2021), 92. https://doi.org/10.1007/s13398-021-01039-x |
| [7] |
W. Chu, Alternating circular sums of binomial coefficients, Bull. Aust. Math. Soc., 106 (2022), 385–395. https://doi.org/10.1017/S0004972722000351 doi: 10.1017/S0004972722000351
|
| [8] | I. M. Gessel, I. Kar, Binomial convolutions for rational power series, arXiv, 2023. https://doi.org/10.48550/arXiv.2304.10426 |
| [9] | S. Janson, Resultant and discriminant of polynomials, Preprint, 2007. Available from: https://www2.math.uu.se/svantejs/papers/sjN5.pdf. |
| [10] | I. Kar, A new method to compute the Hadamard product of two rational functions, Rose-Hulman Undergrad. Math. J., 23 (2022), 3. |
| [11] | T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley & Sons, 2001. https://doi.org/10.1002/9781118033067 |
| [12] | D. E. Knuth, Bracket notation for the "coefficient of operator", In: A. W. Roscoe, A classical mind: essays in honour of C. A. R. Hoare, Hertfordshire: Prentice Hall International Ltd., 1994,247–258. |
| [13] | J. Mikic, A proof of the curious binomial coefficient identity which is connected with the Fibonacci numbers, Open Access J. Math. Theor. Phys., 1 (2017), 1–7. |
| [14] | V. V. Prasalov, Polynomials, Algorithms and Computation in Mathematics, Vol. 11, Springer, 2004. https://doi.org/10.1007/978-3-642-03980-5 |
| [15] | The On-Line Encyclopedia of Integer Sequences (OEIS), accessed on 12 May 2025. Available from: http://oeis.org/. |
Marta Na Chen, Wenchang Chu. Generating functions for circular sums of binomial products[J]. AIMS Mathematics, 2025, 10(12): 28182-28206. doi: 10.3934/math.20251239
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