Binomial type

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

Abstract: 1. Some Methods for closed form Representation.- 1 Some Methods.- 1.1 Introduction.- 1.2 Contour Integration.- 1.3 Use of Integral Equations.- 1.4 Wheelon's Results.- 1.5 Hypergeometric Functions.- 2 A Tree Search Sum and Some Relations.- 2.1 Binomial Summation.- 2.2 Riordan.- 2.3 Method of Jonassen and Knuth.- 2.4 Method of Gessel.- 2.5 Method of Rousseau.- 2.6 Hypergeometric Form.- 2.7 Snake Oil Method.- 2.8 Some Relations.- 2.9 Method of Sister Celine.- 2.10 Method of Creative Telescoping.- 2.11 WZ Pairs Method.- 2. Non-Hypergeometric Summation.- 1 Introduction.- 2 Method.- 3 Burmann's Theorem and Application.- 4 Differentiation and Integration.- 5 Forcing Terms.- 6 Multiple Delays, Mixed and Neutral Equations.- 7 Bruwier Series.- 8 Teletraffic Example.- 9 Neutron Behaviour Example.- 10 A Renewal Example.- 11 Ruin Problems in Compound Poisson Processes.- 12 A Grazing System.- 13 Zeros of the Transcendental Equation.- 14 Numerical Examples.- 15 Euler'sWork.- 16 Jensen's Work.- 17 Ramanujan's Question.- 18 Cohen's Modification and Extension.- 19 Conolly's Problem.- 3. Burmann's Theorem.- 1 Introduction.- 2 Burmann's Theorem and Proof.- 2.1 Applying Burmann's Theorem.- 2.2 The Remainder.- 3 Convergence Region.- 3.1 Extension of the Series.- 4. Binomial type Sums.- 1 Introduction.- 2 Problem Statement.- 3 A Recurrence Relation.- 4 Relations Between Gk (m) and Fk+1 (m).- 5. Generalization of the Euler Sum.- 1 Introduction.- 2 1-Dominant Zero.- 2.1 The System.- 2.2 QR,k (0) Recurrences and Closed Forms.- 2.3 Lemma and Proof of Theorem 5.1.- 2.4 Extension of Results.- 2.5 Renewal Processes.- 3 The K-Dominant Zeros Case.- 3.1 The k-System.- 3.2 Examples.- 3.3 Extension.- 6. Hypergeometric Summation: Fibonacci and Related Series.- 1 Introduction.- 2 The Difference-Delay System.- 3 The Infinite Sum.- 4 The Lagrange Form.- 5 Central Binomial Coefficients.- 5.1 Related Results.- 6 Fibonacci, Related Polynomials and Products.- 7 Functional Forms.- 7. Sums and Products of Binomial Type.- 1 Introduction.- 2 Technique.- 3 Multiple Zeros.- 4 More Sums.- 5 Other Forcing Terms.- 8. Sums of Binomial Variation.- 1 Introduction.- 2 One Dominant Zero.- 2.1 Recurrences.- 2.2 Proof of Conjecture.- 2.3 Hypergeometric Functions.- 2.4 Forcing Terms.- 2.5 Products of Central Binomial Coefficients.- 3 Multiple Dominant Zeros.- 3.1 The k Theorem.- 4 Zeros.- 4.1 Numerical Results and Special Cases.- 4.2 The Hypergeometric Connection.- 5 Non-zero Forcing Terms.- References.- About the Author.