INCOMPLETE AND COMPLETE r-CENTRAL LAH-BELL POLYNOMIALS
0
조회
0
다운로드
1
인용
0
참고문헌
키워드
초록
The complete and incomplete Bell polynomials are respectively multivariate forms for Bell polynomials and Stirling numbers of the second kind. These polynomials play a crucial role in enumerative combinatorics. Recently, H.K. Kim introduced both the central Lah numbers and r- central Lah numbers. With this in mind, this paper is subdivided into two parts to address the above mathematical concepts. In the first part, we introduce both complete and incomplete central Lah-Bell polynomials, as multivariate forms of the central Lah numbers and the central Lah-Bell polynomi- als, respectively. We give a relation between the central Lah-Bell numbers and the complete Bell polynomials by using the Kolbig-Coeffey equation. This entails deriving explicit formulas for these polynomials and numbers, as well as conducting research into some identities for these polynomials. In the second part, both the r-central complete and incomplete Lah-Bell polynomials, as multivariate forms of both the r-central Lah-numbers and the r-extended Lah-Bell polynomials, are introduced and explicit formulas and several noble identities are also derived.
Abstract
The complete and incomplete Bell polynomials are respectively multivariate forms for Bell polynomials and Stirling numbers of the second kind. These polynomials play a crucial role in enumerative combinatorics. Recently, H.K. Kim introduced both the central Lah numbers and r- central Lah numbers. With this in mind, this paper is subdivided into two parts to address the above mathematical concepts. In the first part, we introduce both complete and incomplete central Lah-Bell polynomials, as multivariate forms of the central Lah numbers and the central Lah-Bell polynomi- als, respectively. We give a relation between the central Lah-Bell numbers and the complete Bell polynomials by using the Kolbig-Coeffey equation. This entails deriving explicit formulas for these polynomials and numbers, as well as conducting research into some identities for these polynomials. In the second part, both the r-central complete and incomplete Lah-Bell polynomials, as multivariate forms of both the r-central Lah-numbers and the r-extended Lah-Bell polynomials, are introduced and explicit formulas and several noble identities are also derived.
참고문헌 (32)
- Bell, E.T., "Exponenential polynomials", Ann. Math, 35, pp.258~277 (1934)
- Broder, A.Z, "The r-Stirling numbers", Discrete Math, 49(3), pp.241~259 (1984)
- Butzer, P.L, "Central factorial numbers;their main properties and some applications", Numer.Funct.Anal Optim, 10(5-6), pp.419~488 (1989)
- Charalambides, C.A, "Central factorial numbers and related expansions", Fibonacci Quart, 19, pp.451~456 (1981)
- Coffet, M.W., "A set of identites for a class of alternating binomial sums arising in computing applications", arXiv.math-ph
- Comtet, L., "Advanced Combinatorics: The Art of Finite and Infinite Expansions" (1974), Reidel
- Connon, D.F, "Various applications of the (exponential) complete Bell polynomials", arXiv:1001.2835, ~ (2010)
- Cvijove ´ c, D, "New identities for the partial Bell polynomials", Appl.Math.Mon.Lett, 24, pp.1544~1547 (2011)
- 김대산, "Lah-Bell numbers and polynomials", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 23(4), pp.577~586 (2020), 장전수학회
- 김태균, "ON CENTRAL FUBINI POLYNOMIALS ASSOCIATED WITH CENTRAL FACTORIAL NUMBERS OF THE SECOND KIND", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 21(4), pp.589~598 (2018), 장전수학회[DOI]
- Kim, D. S., "Some identities on r-central factorial numbers and r-central Bell polynomials", Adv. Difference Equ, 2019, pp.245~ (2019)
- Kim, D.S, "On r-central incomplete and complete Bell polynomials", Symmetry, 11, pp.724~735 (2019)
- Kim, H.K, "Central Lah numbers and central Lah-Bell numbers", ~
- im, H.K, "Note on the central r-Lah numbers and the central r-Lah-Bell numbers", AIMS Mathematics, 7(2), pp.2929~2939
- 김혜경, "NOTE ON EXTENDED LAH-BELL POLYNOMIALS AND DEGENERATE EXTENDED LAH-BELL POLYNOMIALS", Advanced Studies in Contemporary Mathematics, 30(4), pp.547~558 (2020), 장전수학회
- 김대산, "r-EXTENDED LAH-BELL NUMBERS AND POLYNOMIALS ASSOCIATED WITH r-LAH NUMBERS", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 24(1), pp.1~10 (2021), 장전수학회
- 김태균, "A NOTE ON CENTRAL FACTORIAL NUMBERS", Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집), 21(4), pp.575~588 (2018), 장전수학회[DOI]
- Kim, T, "On degenerate central complete Bell polynomials", Appl.Anal. Discrete Math, 13(3), pp.805~818 (2019)
- Kim, T., "A note on central Bell numbers and polynomials", Russ. J. Math. Phys, 27(1), pp.76~81 (2020)
- Kim, T., "Generalized degenerate Bernoulli numbers and polynomials arising from Gauss hypergeometric function", Adv. Difference Equ, pp.175~186 (2021)
- Kim, T., "Complete and incomplete Bell polynomials associated with Lah-Bell numbers and polynomials", Adv. Difference Equ, pp.101~112 (2021)
- K¨olbig, K.S., "The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind", J. Comput. Appl. Math, 51, pp.113~116 (1994)
- Lah, I., "A new kind of numbers and its application in the actuarial mathematics", Bol. Inst. Acyuar. Port, 9, pp.7~15 (1954)
- Martinjak, I, "Lah numbers and Lindstrom’s lemma", C. R. Acad. Sci. Paris, Ser.I, 356, pp.5~7 (2018)
- Mihoubi, M, "Bell polynomials and binomial type sequences", Discret. Math, 308, pp.2450~2459 (2008)
- Mihoubi, M., "The partial r-Bell polynomials", Afr. Mat (2017)[DOI]
- Nyul, G., "The r-Lah numbers", Discrete Math, 338, pp.1660~1666 (2015)[DOI]
- Nyul, G, "Sums of r-Lah numbers and r-Lah polynomials", ARS Math. contemp (2020)[DOI]
- Port, D, "Polynomial maps with applications to combinatorics and probability theory", Ph.D, pp.1994
- Riordan, J., "Combinatorial Identities" (1968), John Wiley and Sons, Inc
- Ramirez, C., "A (p, q)-analogue of the r-Whitney-Lah numbers", J. Integer Seq, 19, pp.16.5.6~ (2016)
- Steffensen, J. F, "Interpolation" (1927)