Researchers Blend Probability and Pattern Theory in New Study

:::info Authors:

(1) Yuankui Ma, School of Science, Xi’An Technological University, Xi’An 710021, Shaanxi, China ([email protected]);

(2) Taekyun Kim, School of Science, Xi’An Technological University, Xi’An 710021, Shaanxi, China; Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea ([email protected]);

(3) Dae San Kim. Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea ([email protected]).

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Table of Links

Abstract and 1. Introduction

1. Probabilistic LAH Numbers and LAH-BELL Polynomials
2. Conclusion and References

\ ABSTRACT. Let Y be a random variable whose moment generating function exists in some neighborhood of the origin. The aim of this paper is to study the probabilistic Lah numbers associated with Y and the probabilistic Lah-Bell polynomials associated with Y, as probabilistic versions of the Lah numbers and the Lah-Bell polynomials, respectively. We derive some properties, explicit expressions, recurrence relations and certain identities for those numbers and polynomials. In addition, we treat the special cases that Y is the Poisson random variable with parameter α > 0 and the Bernoulli random variable with probability of success p.

1. INTRODUCTION

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\ We recall that the falling factorial sequence is defined by

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2. PROBABILISTIC LAH NUMBERS AND LAH-BELL POLYNOMIALS

\ Therefore, by comparing the coefficients on both sides of (16), we obtain the following theorem.

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\ From (15), (18), we note that

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\ Therefore, by (19), we obtain the following theorem.

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\ Therefore, by comparing the coefficients on both sides of (24), we obtain the following theorem.

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\ Therefore, by (27), we obtain the following theorem.

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3. CONCLUSION

\ As one of our future projects, we would like to continue to investigate degenerate versions, λanalogues and probabilistic versions of many special polynomials and numbers and to find their applications to physics, science and engineering as well as to mathematics.

REFERENCES

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