SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS

Feng Qi; Xiao-Ting Shi; Fang-Fang Liu; Dmitry V. Kruchinin

doi:10.11948/2017054

2017 Volume 7 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(3): 857-871

Next Article Previous Article

Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, Dmitry V. Kruchinin. SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 857-871. doi: 10.11948/2017054

Citation:

Feng Qi, Xiao-Ting Shi, Fang-Fang Liu, Dmitry V. Kruchinin. SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 857-871. doi: 10.11948/2017054

SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS

1 Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China;
2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China;
3 Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China;
4 Tomsk Polytechnic University, 634050, Tomsk, Russia;
5 Tomsk State University of Control Systems and Radioelectronics, Tomsk, 634050, Russia

Abstract

Abstract

In the paper, the authors establish several explicit formulas for special values of the Bell polynomials of the second kind, connect these formulas with the Bessel polynomials, and apply these formulas to give new expressions for the Catalan numbers and to compute arbitrary higher order derivatives of elementary functions such as the since, cosine, exponential, logarithm, arcsine, and arccosine of the square root for the variable.
- Bell polynomial of the second kind /
- formula /
- Catalan number /
- connection /
- Bessel polynomial /
- derivative /
- elementary function
MSC: 05A10;11B75;11B83;11C08;26A09;26A24;33B10;33B99

FullText(HTML)

References

[1]	L. Comtet, Advanced Combinatorics:The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., 1974. Google Scholar
[2]	R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics\|A Foundation for Computer Science, 2nd ed., Addison-Wesley Publishing Company, Reading, MA, 1994. Google Scholar
[3]	T. Koshy, Catalan Numbers with Applications, Oxford University Press, Oxford, 2009. Google Scholar
[4]	H. L. Krall and O. Frink, A new class of orthogonal polynomials:The Bessel polynomials, Trans. Amer. Math. Soc., 1949, 65, 100-115. Google Scholar
[5]	V. V. Kruchinin and D. V. Kruchinin, Composita and its properties, J. Anal. Number Theory, 2014, 2(2), 37-44. Google Scholar
[6]	W. Lang, On polynomials related to derivatives of the generating function of Catalan numbers, Fibonacci Quart., 2002, 40(4), 299-313. Google Scholar
[7]	W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fibonacci Quart., 2000, 38(5), 408-419. Google Scholar
[8]	A. Nkwanta and A. Tefera, Curious relations and identities involving the Catalan generating function and numbers, J. Integer Seq., 2013, 16(9), Article 13.9.5, 15 pages. Google Scholar
[9]	F. Qi, X.-T. Shi, M. Mahmoud, and F.-F. Liu, The Catalan numbers:a generalization, an exponential representation, and some properties, J. Comput. Anal. Appl., 2017, 23(5), 937-944. Google Scholar
[10]	I. Vardi, Computational Recreations in Mathematica, Addison-Wesley, Redwood City, CA, 1991. Google Scholar
[11]	WikiPedia, Bessel polynomials, From the Free Encyclopedia. Available online at https://en.wikipedia.org/wiki/Bessel_polynomials. Google Scholar
[12]	WikiPedia, Catalan number, From the Free Encyclopedia. Available online at https://en.wikipedia.org/wiki/Catalan_number. Google Scholar