Differential and Difference Equations A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday Printed Edition of the Special Issue Published in Axioms www.mdpi.com/journal/axioms Sotiris K. Ntouyas Edited by Differential and Difference Equations Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday Editor Sotiris K. Ntouyas MDPI • Basel • Beijing • Wuhan • Barcelona • Belgrade • Manchester • Tokyo • Cluj • Tianjin Editor Sotiris K. Ntouyas University of Ioannina Greece Editorial Office MDPI St. Alban-Anlage 66 4052 Basel, Switzerland This is a reprint of articles from the Special Issue published online in the open access journal Axioms (ISSN 2075-1680) (available at: https://www.mdpi.com/journal/axioms/special issues/ differ equ). For citation purposes, cite each article independently as indicated on the article page online and as indicated below: LastName, A.A.; LastName, B.B.; LastName, C.C. Article Title. Journal Name Year , Article Number , Page Range. ISBN 978-3-03943-068-0 ( H bk) ISBN 978-3-03943-069-7 (PDF) c © 2020 by the authors. Articles in this book are Open Access and distributed under the Creative Commons Attribution (CC BY) license, which allows users to download, copy and build upon published articles, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. The book as a whole is distributed by MDPI under the terms and conditions of the Creative Commons license CC BY-NC-ND. Contents About the Editor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Preface to ”Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday” . . . . . . . . . . . . . . . . . . . . . . . . . ix Irem Kucukoglu, Burcin Simsek and Yilmaz Simsek Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function Reprinted from: Axioms 2019 , 8 , 112, doi:10.3390/axioms8040112 . . . . . . . . . . . . . . . . . . 1 Temirkhan Aleroev On One Problems of Spectral Theory for Ordinary Differential Equations of Fractional Order Reprinted from: Axioms 2019 , 8 , 117, doi:10.3390/axioms8040117 . . . . . . . . . . . . . . . . . . 17 Rasool Shah, Hassan Khan, Dumitru Baleanu Fractional Whitham–Broer–Kaup Equations within Modified Analytical Approaches Reprinted from: Axioms 2019 , 8 , 125, doi:10.3390/axioms8040125 . . . . . . . . . . . . . . . . . . . 25 Paolo Emilio Ricci and Pierpaolo Natalini General Linear Recurrence Sequences and Their Convolution Formulas Reprinted from: Axioms 2019 , 8 , 132, doi:10.3390/axioms8040132 . . . . . . . . . . . . . . . . . . 47 Chenkuan Li and Hunter Plowman Solutions of the Generalized Abel’s Integral Equationsof the Second Kind with Variable Coefficients Reprinted from: Axioms 2019 , 8 , 137, doi:10.3390/axioms8040137 . . . . . . . . . . . . . . . . . . 59 Nak Eun Cho and Jacek Dziok Harmonic Starlike Functions with Respect to Symmetric Points Reprinted from: Axioms 2020 , 9 , 3, doi:10.3390/axioms9010003 . . . . . . . . . . . . . . . . . . . . 69 Choonkil Park, Osama Moaaz and Omar Bazighifan Oscillation Results for Higher Order Differential Equations Reprinted from: Axioms 2020 , 9 , 14, doi:10.3390/axioms9010014 . . . . . . . . . . . . . . . . . . . 81 Leonid Shaikhet Stability of Equilibria of Rumor Spreading Model under Stochastic Perturbations Reprinted from: Axioms 2020 , 9 , 24, doi:10.3390/axioms9010024 . . . . . . . . . . . . . . . . . . . 93 Nita H Shah, Nisha Sheoran and Yash Shah Dynamics of HIV-TB Co-Infection Model Reprinted from: Axioms 2020 , 9 , 29, doi:10.3390/axioms9010029 . . . . . . . . . . . . . . . . . . . 109 Abdeljabbar Talal Yousef and Zabidin Salleh On a Harmonic Univalent Subclass of Functions Involving a Generalized Linear Operator Reprinted from: Axioms 2020 , 9 , 32, doi:10.3390/axioms9010032 . . . . . . . . . . . . . . . . . . . 123 Donal O’Regan Coincidence Continuation Theory for Multivalued Maps with Selections in a Given Class Reprinted from: Axioms 2020 , 9 , 37, doi:10.3390/axioms9020037 . . . . . . . . . . . . . . . . . . . 133 v Rabha W. Ibrahim, Rafida M. Elobaid and Suzan J. Obaiys Generalized Briot-Bouquet Differential Equation Based on New Differential Operator with Complex Connections Reprinted from: Axioms 2020 , 9 , 42, doi:10.3390/axioms9020042 . . . . . . . . . . . . . . . . . . . 141 Tursun K. Yuldashev Nonlocal Inverse Problem for a Pseudohyperbolic- PseudoellipticType Integro-Differential Equations Reprinted from: Axioms 2020 , 9 , 45, doi:10.3390/axioms9020045 . . . . . . . . . . . . . . . . . . . 155 Davor Dragiˇ cevi ́ c and Ciprian Preda Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiflows with Discrete Time Reprinted from: Axioms 2020 , 9 , 47, doi:10.3390/axioms9020047 . . . . . . . . . . . . . . . . . . . 177 Ahmed Alsaedi, Abrar Broom, Sotiris K. Ntouyas and Bashir Ahmad Nonlocal Fractional Boundary Value Problems Involving Mixed Right and Left Fractional Derivatives and Integrals Reprinted from: Axioms 2020 , 9 , 50, doi:10.3390/axioms9020050 . . . . . . . . . . . . . . . . . . . 189 R. Leelavathi, G. Suresh Kumar, Ravi P. Agarwal, Chao Wang, and M.S.N. Murty Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales Reprinted from: Axioms 2020 , 9 , 65, doi:10.3390/axioms9020065 . . . . . . . . . . . . . . . . . . . 205 Bashir Ahmad, Najla Alghamdi, Ahmed Alsaedi and Sotiris K. Ntouyas Existence Results for Nonlocal Multi-Point and Multi-Term Fractional Order Boundary Value Problems Reprinted from: Axioms 2020 , 9 , 70, doi:10.3390/axioms9020070 . . . . . . . . . . . . . . . . . . . 229 Ilya Boykov, Vladimir Roudnev and Alla Boykova Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations Reprinted from: Axioms 2020 , 9 , 74, doi:10.3390/axioms9030074 . . . . . . . . . . . . . . . . . . . 251 Osama Moaaz, Hamida Mahjoub and Ali Muhib On the Periodicity of General Class of Difference Equations Reprinted from: Axioms 2020 , 9 , 75, doi:10.3390/axioms9030075 . . . . . . . . . . . . . . . . . . . 269 Osama Moaaz , Hamida Mahjoub and Ali Muhib Eigenfunction Families and Solution Bounds forMultiplicatively Advanced Differential Equations Reprinted from: Axioms 2020 , 9 , 83, doi:10.3390/axioms9030083 . . . . . . . . . . . . . . . . . . . 281 Alexander Yeliseev On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator Reprinted from: Axioms 2020 , 9 , 86, doi:10.3390/axioms9030086 . . . . . . . . . . . . . . . . . . . 309 Konstantinos Kalimeris and Athanassios S. Fokas The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem Reprinted from: Axioms 2020 , 9 , 89, doi:10.3390/axioms9030089 . . . . . . . . . . . . . . . . . . . 327 Thomas Ernst On the Triple Lauricella–Horn–Karlsson q -Hypergeometric Functions Reprinted from: Axioms 2020 , 9 , 93, doi:10.3390/axioms9030093 . . . . . . . . . . . . . . . . . . . 345 vi About the Editor Sotiris K. Ntouyas is Professor Emeritus in the Department of Mathematics of the University of Ioannina, Greece. He received his BS degree and PhD from the University of Ioannina, in 1972 and 1980, respectively. His research interests include initial and boundary value problems for differential equations (ordinary, functional, with deviating arguments, neutral, partial, integrodifferential, inclusions, impulsive, fuzzy, stochastic, fractional), inequalities, asymptotic behavior and controllability. More than 640 of his papers have appeared in print or have been accepted for publication in refereed journals. He is the co-author of the books Impulsive differential equations and inclusions, Controllabibility for semilinear functional differential equations and inclusions, Quantum Calculus: New Concepts, Impulsive IVPs and BVPs, Inequalities and Hadamard-Type Fractional Differential Equations, Inclusions and Inequalities . He is a member of 21 international journals’ Editorial Boards, and a reviewer for many international journals. He appears in the 2018 list, published by Clarivate Analytics, of Highly Cited Researchers. vii Preface to ”Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday” It is our great pleasure to publish this book. All contents were peer-reviewed by multiple referees and published as papers in the Special Issue ”Differential and Difference Equations: A Themed Issue Dedicated to Prof. Hari M. Srivastava on the Occasion of his 80th Birthday” in the journal Axioms. These studies provide new and interesting results in different branches of differential equations, so that the readers will be able to obtain the latest developments in the fields of differential equations. Sotiris K. Ntouyas Editor ix axioms Article Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson–Charlier Polynomials and Probability Distribution Function Irem Kucukoglu 1 , Burcin Simsek 2 and Yilmaz Simsek 3, * 1 Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, TR-07425 Antalya, Turkey; irem.kucukoglu@alanya.edu.tr 2 Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA; bus5@pitt.edu 3 Department of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya, Turkey * Correspondence: ysimsek@akdeniz.edu.tr Received: 14 September 2019; Accepted: 9 October 2019; Published: 11 October 2019 Abstract: The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p -adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. Keywords: generating functions; functional equations; partial differential equations; special numbers and polynomials; Bernoulli numbers; Euler numbers; Stirling numbers; Bell polynomials; Cauchy numbers; Poisson-Charlier polynomials; Bernstein basis functions; Daehee numbers and polynomials; combinatorial sums; binomial coefficients; p -adic integral; probability distribution MSC: Primary 05A10; 05A15; 11B73; 11B68; 11B83; Secondary 05A19; 11B37; 11S23; 26C05; 34A99; 35A99; 40C10 1. Introduction In recent years, generating functions and their applications on functional equations and differential equations has gained high attention in various areas. These techniques allow researchers to derive various identities and combinatorial sums that yield important special numbers and polynomials. In fact, the current trend is to combine the p -adic integrals with these techniques. In most of fields of mathematics and physics, different applications of generating functions are used as an important tool. For instance, a common research topic in quantum physics is to identify a generating function that could be a solution to a differential equation. The motivation of this paper is to outline the advantages of techniques associated with generating functions. First, generating functions are presented for new families of combinatorial numbers and polynomials. Second, we derive new identities, relations, and formulas including the Bersntein Axioms 2019 , 8 , 112; doi:10.3390/axioms8040112 www.mdpi.com/journal/axioms 1 Axioms 2019 , 8 , 112 basis functions, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, the Daehee numbers and polynomials, the probability distribution functions, as well as combinatorial sums including the Bernoulli numbers, the Euler numbers, the Cauchy numbers (or the Bernoulli numbers of the second kind), and combinatorial numbers. With the followings, we briefly introduce the notations, definitions, relations, and formulas are used throughout this paper: As usual, let N , Z , N 0 , Q , R , and C denote the set of natural numbers, set of integers, set of nonnegative integers, set of rational numbers, set of real numbers, and set of complex numbers, respectively. Let log z denote the principal branch of the multi-valued function log z with the imaginary part Im ( log z ) constrained by the interval ( − π , π ] . We also assume that: 0 n = [ 1, if n = 0 0, if n ∈ N Moreover, ] z v ) = z ( z − 1 ) · · · ( z − v + 1 ) v ! = ( z ) v v ! ( v ∈ N , z ∈ C ) so that, ] z 0 ) = ( z ) 0 = 1 ( cf . [1–31]). The Poisson–Charlier polynomials C n ( x ; a ) , which are members of the family of Sheffer-type sequences, are defined as below: F pc ( t , x ; a ) = e − t ] t a + 1 ) x = ∞ ∑ n = 0 C n ( x ; a ) t n n ! , (1) where, C n ( x ; a ) = n ∑ j = 0 ( − 1 ) n − j ] n j ) ( x ) j a j , (2) ( cf . [16], (p. 120, [18]), [24]). Let x ∈ [ 0, 1 ] and let n and k be nonnegative integers. The Bernstein basis functions, B n k ( x ) , are defined by: B n k ( x ) = ] n k ) x k ( 1 − x ) n − k , ( k = 0, 1, . . . , n ) (3) so that, ] n k ) = n ! k ! ( n − k ) ! and its generating function is given by: F B ( t , x ; k ) = ( xt ) k e ( 1 − x ) t k ! = ∞ ∑ n = 0 B n k ( x ) t n n ! , (4) where t ∈ C ( cf . [1,15,20,26]). The Stirling numbers of the first kind, S 1 ( n , k ) , are defined by the following generating function: F S 1 ( t ; k ) = ( log ( 1 + t )) k k ! = ∞ ∑ n = k S 1 ( n , k ) t n n ! , ( k ∈ N 0 ) (5) 2 Axioms 2019 , 8 , 112 so that, ( x ) n = n ∑ k = 0 S 1 ( n , k ) x k (6) ( cf . [2–4,29,30]; see also the references cited therein). The λ -Stirling numbers of the second kind, S 2 ( n , k ; λ ) , are defined with generating function given below ( cf . [21,30]): F S 2 ( t ; v ; λ ) = ( λ e t − 1 ) v v ! = ∞ ∑ n = 0 S 2 ( n , v ; λ ) t n n ! , ( v ∈ N 0 ) (7) Notice here that, when λ = 1, this reduces to the Stirling numbers of the second kind, S 2 ( n , v ) , whose generating function is given below: F S 2 ( t ; v ) = ( e t − 1 ) v v ! = ∞ ∑ n = 0 S 2 ( n , v ) t n n ! , ( v ∈ N 0 ) , (8) namely, S 2 ( n , v ) = S 2 ( n , v ; 1 ) ( cf . [2,5,21,30]). The Bell polynomials (i.e., exponential polynomials), Bl n ( x ) , is defined by: Bl n ( x ) = n ∑ v = 1 S 2 ( n , v ) x v (9) so that the generating function for the Bell polynomials is given by: F Bell ( t , x ) = e ( e t − 1 ) x = ∞ ∑ n = 0 Bl n ( x ) t n n ! (10) ( cf . [4,18]). The numbers Y ( k ) n ( λ ) and the polynomials Y ( k ) n ( x ; λ ) are defined by the following generating functions, respectively: F ( t , k ; λ ) = ] 2 λ ( 1 + λ t ) − 1 ) k = ∞ ∑ n = 0 Y ( k ) n ( λ ) t n n ! , (11) and, F ( t , x , k ; λ ) = F ( t , k ; λ ) ( 1 + λ t ) x = ∞ ∑ n = 0 Y ( k ) n ( x ; λ ) t n n ! , (12) where k ∈ N 0 and λ is real or complex number ( cf . [14]). Substituting k = 1 into Equation (11), we have: Y n ( λ ) = Y ( 1 ) n ( λ ) ( cf . [23]). Substituting k = 1 and λ = − 1 into Equation (11), we get the following well-known relation between the numbers Y n ( λ ) and the Changhee numbers of the first kind, Ch n : Ch n = ( − 1 ) n + 1 Y n ( − 1 ) Thus we have, Ch n = ( − 1 ) n n ! n + 1 = n ∑ k = 0 S 1 ( n , k ) E k (13) 3 Axioms 2019 , 8 , 112 where the Changhee numbers of the first kind, Ch n are defined means of the following generating function: 2 t + 1 = ∞ ∑ n = 0 Ch n t n n ! (14) ( cf . [9], see also [7]). The Daehee polynomials, D n ( x ) , is defined by the following generating functions ( cf . [8]): F D ( x , t ) = log ( 1 + t ) t ( 1 + t ) x = ∞ ∑ n = 0 D n ( x ) t n n ! (15) which, for x = 0, corresponds the generating functions of the Daehee number, D n = D n ( 0 ) , given by the following explicit formula: D n = ( − 1 ) n n ! n + 1 (16) The combinatorial numbers, y 1 ( n , k ; λ ) , are defined by the following generating function: F y 1 ( t , k ; λ ) = 1 k ! ( λ e t + 1 ) k = ∞ ∑ n = 0 y 1 ( n , k ; λ ) t n n ! (17) where k ∈ N 0 and λ ∈ C ( cf . [22]). Use the preceding generating function for the combinatorial numbers, y 1 ( n , k ; λ ) to compute the following explicit formula: y 1 ( n , k ; λ ) = 1 k ! k ∑ j = 0 ] k j ) λ j j n (18) ( cf . [22] (Theorem 1, Equation (9))). Note that the following equality holds true: y 1 ( n , k ; λ ) = 1 k ! d n dt n ( λ e t + 1 ) k ∣ ∣ ∣ ∣ t = 0 (19) ( cf . [31] (p. 64)). When λ = 1, if we multiply the numbers y 1 ( n , k ; λ ) by k ! , then Equation (18) is reduced to the following combinatorial numbers ( cf . [6,19,22]): B ( k , n ) = k ∑ j = 0 ] k j ) j n which satisfies the following differential equation: B ( k , n ) = d n dt n ( e t + 1 ) k ∣ ∣ ∣ ∣ t = 0 (20) ( cf . [6], (Equation (2), p. 2 [22])). The combinatorial numbers B ( n , k ) have various kinds of combinatorial applications. For instance, Ross [ 19 ] (pp. 18–20, Exercises 10–12) gave the following applications for solutions of exercises 10–12: From a group of n people, suppose that we want to choose a committee of k , k ≤ n , one of whom is to be designated as chairperson. How many different selections are there in which the chairperson and the secretary are the same? Ross [19] (p. 18, Exercise 12) gave the following answer: B ( 1, n ) = n 2 n − 1 4 Axioms 2019 , 8 , 112 By using the preceding idea summarized above, the following combinatorial identities are obtained: n ∑ k = 0 ] n k ) k = n 2 n − 1 n ∑ k = 0 ] n k ) k 2 = 2 n − 2 n ( n + 1 ) and, n ∑ k = 0 ] n k ) k 3 = 2 n − 3 n 2 ( n + 3 ) ( cf (pp. 18–20, Exercises 10–12 [ 19 ]), [ 22 , 25 ]). Observe that these numbers are also arised from Equation (20). Next, we present the outline of the present paper: In Section 2, we construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions, we not only investigate properties of these new families, but also provide some new identities and relations with the inclusion of the Bersntein basis functions, combinatorial numbers, and the Stirling numbers. In Section 3, we obtain some derivative formulas and recurrence relations for these new families of combinatorial numbers and polynomials by using differential equations that are a result of these generating functions and their partial derivatives. In Section 4, by using functional equations of the generating functions, we derive some formulas and combinatorial sums including binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson–Charlier polynomials, combinatorial numbers and polynomials, and the Bersntein basis functions. In Section 5, by applying the p -adic integrals and Riemann integral to some new formulas derived by the authors of this paper, some combinatorial sums comprising the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e. exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind) are presented. In Section 6, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution. In Section 7, we conclude our findings. 2. New Families of the Combinatorial Numbers and Polynomials In this section, we define new families of the combinatorial numbers and polynomials by the following generating functions, respectively: G ( t , k ; λ ) = 2 − k ( λ ( 1 + λ t ) − 1 ) k = ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! (21) and, G ( t , x , k ; λ ) = G ( t , k ; λ ) ( 1 + λ t ) x = ∞ ∑ n = 0 Q n ( x ; λ , k ) t n n ! (22) where k ∈ N and λ is a real or complex number. Combining Equations (21) and (22), we get: ∞ ∑ n = 0 Q n ( x ; λ , k ) t n n ! = ∞ ∑ n = 0 n ∑ j = 0 ] n j ) λ n − j Y ( − k ) j ( λ ) ( x ) n − j t n n ! . (23) Comparing coefficient of t n n ! on both sides of the above equation, we arrive at the following theorem: 5 Axioms 2019 , 8 , 112 Theorem 1. Q n ( x ; λ , k ) = n ∑ j = 0 ] n j ) λ n − j Y ( − k ) j ( λ ) ( x ) n − j (24) By the binomial theorem, we have: ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! = 2 − k ∞ ∑ n = 0 ] k n ) λ 2 n ( λ − 1 ) k − n t n Comparing the coefficient of t n on both sides of the above equation, we arrive at the following theorem: Theorem 2. Let k and n be nonnegative integers. Then: Y ( − k ) n ( λ ) = [ 2 − k n ! ( k n ) λ 2 n ( λ − 1 ) k − n if n ≤ k 0 if n > k (25) By Equation (25), a few values of the numbers Y ( − k ) n ( λ ) are computed as follows: Y ( − k ) 0 ( λ ) = 2 − k ( λ − 1 ) k , Y ( − k ) 1 ( λ ) = 2 − k ] k 1 ) λ 2 ( λ − 1 ) k − 1 , Y ( − k ) 2 ( λ ) = 2 − k 2! ] k 2 ) λ 4 ( λ − 1 ) k − 2 , Y ( − k ) j ( λ ) = 2 − k j ! ] k j ) λ 2 j ( λ − 1 ) k − j for j ≤ k , Y ( − k ) k ( λ ) = 2 − k k ! λ 2 k , Y ( − k ) j ( λ ) = 0 for j > k By Equations (24) and (25), we also compute a few values of the polynomials Q n ( x ; λ , k ) as follows: Q 0 ( x ; λ , k ) = 2 − k ( λ − 1 ) k , Q 1 ( x ; λ , k ) = 2 − k ( λ − 1 ) k λ x + 2 − k k λ 2 ( λ − 1 ) k − 1 , Q 2 ( x ; λ , k ) = 2 − k ( λ − 1 ) k λ 2 x 2 + ( − 2 − k ( λ − 1 ) k λ 2 + 2 − k + 1 k λ 3 ( λ − 1 ) k − 1 ) x + 2 − k k ( k − 1 ) λ 4 ( λ − 1 ) k − 1 By Equation (3), we arrive at a computation formula, for the numbers Y ( − k ) n ( λ ) , in terms of the Bernstein basis functions by the following corollary: Corollary 1. Let n and k be nonnegative integers and λ ∈ [ 0, 1 ] . Then, Y ( − k ) n ( λ ) = [ 2 − k n ! ( − 1 ) k − n λ n B k n ( λ ) if n ≤ k 0 if n > k (26) 6 Axioms 2019 , 8 , 112 Replacing 1 + λ t by e log ( 1 + λ t ) leads Equation (21) to be: ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! = ( − 1 ) k 2 k ( − λ e log ( 1 + λ t ) + 1 ) k (27) By combining Equation (17) with the above quation, we get: ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! = ( − 1 ) k k ! 2 k ∞ ∑ m = 0 y 1 ( m , k ; − λ ) ( log ( 1 + λ t )) m m ! (28) which follows from Equation (5) that: ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! = ( − 1 ) k k ! 2 k ∞ ∑ n = 0 n ∑ m = 0 λ n y 1 ( m , k ; − λ ) S 1 ( n , m ) t n n ! . (29) Therefore, by comparing coefficient of t n n ! on both sides of the above equation, we arrive at the following theorem: Theorem 3. Y ( − k ) n ( λ ) = ( − 1 ) k k ! 2 k n ∑ m = 0 λ n y 1 ( m , k ; − λ ) S 1 ( n , m ) (30) Combining Equations (25) with (30) yields the following corollary: Corollary 2. n ∑ m = 0 y 1 ( m , k ; − λ ) S 1 ( n , m ) = [ ( − 1 ) k λ n ( λ − 1 ) k − n ( k − n ) ! if n ≤ k 0 if n > k (31) If we also combine Equations (26) with (30), then we have the following result: Corollary 3. Let n and k nonnegative integer with n ≤ k. Then, n ∑ m = 0 y 1 ( m , k ; − λ ) S 1 ( n , m ) = ( − 1 ) n n ! k ! B k n ( λ ) (32) On the other hand, since the following equality holds true ( cf . [13]): S 2 ( n , k ; λ ) = ( − 1 ) k y 1 ( n , k ; − λ ) , (33) Equation (31) leads the following corollary: Corollary 4. n ∑ m = 0 S 2 ( m , k ; λ ) S 1 ( n , m ) = [ λ n ( λ − 1 ) k − n ( k − n ) ! if n ≤ k 0 if n > k (34) 3. Derivative Formulas and Recurrence Relations Arising from Differential Equations of Generating Functions In this section, by using differential equations involving the generating functions G ( t , k ; λ ) and G ( t , x , k ; λ ) and their partial derivatives with respect to the parameters t , λ , and x , we obtain some derivative formulas and recurrence relations for the numbers Y ( − k ) n ( λ ) and the polynomials Q n ( x ; λ , k ) 7 Axioms 2019 , 8 , 112 Differentiating both sides of Equation (21) with respect to λ , we get the following partial derivative equation: ∂ ∂λ {G ( t , k ; λ ) } = k 2 ( 2 λ t + 1 ) G ( t , k − 1; λ ) (35) Also, if we differentiate both sides of Equation (21) with respect to t , then we get the following partial derivative equation: ∂ ∂ t {G ( t , k ; λ ) } = k λ 2 2 G ( t , k − 1; λ ) (36) By combining Equation (35) with the RHS of Equation (21), we obtain: ∞ ∑ n = 0 d d λ { Y ( − k ) n ( λ ) } t n n ! = k 2 ∞ ∑ n = 0 ( 2 n λ Y ( − k + 1 ) n − 1 ( λ ) + Y ( − k + 1 ) n ( λ ) ) t n n ! . (37) Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the following theorem: Theorem 4. Let n ∈ N . Then, we have: d d λ { Y ( − k ) n ( λ ) } = k 2 ( 2 n λ Y ( − k + 1 ) n − 1 ( λ ) + Y ( − k + 1 ) n ( λ ) ) (38) By combining Equation (36) with the RHS of Equation (21), we get: ∂ ∂ t ∞ ∑ n = 0 Y ( − k ) n ( λ ) t n n ! = k λ 2 2 ∞ ∑ n = 0 Y ( − k + 1 ) n ( λ ) t n n ! . (39) which, by comparing the coefficients of t n n ! on both sides of the above equation, yields the following theorem: Theorem 5. Let n ∈ N 0 . Then, we have: Y ( − k ) n + 1 ( λ ) = k λ 2 2 Y ( − k + 1 ) n ( λ ) (40) Differentiating both sides of Equation (22) with respect to λ , we get the following partial derivative equation: ∂ ∂λ {G ( t , x , k ; λ ) } = k 2 ( 2 λ t + 1 ) G ( t , x , k − 1; λ ) + xt G ( t , x − 1, k ; λ ) (41) Furthermore, if we differentiate both sides of the Equation (22) with respect to t , then we also get the following partial derivative equation: ∂ ∂ t {G ( t , x , k ; λ ) } = k λ 2 2 G ( t , x , k − 1; λ ) + x λ G ( t , x − 1, k ; λ ) (42) Additionally, when we differentiate both sides of Equation (22) with respect to x , we also get the following partial derivative equation: ∂ ∂ x {G ( t , x , k ; λ ) } = log ( 1 + λ t ) G ( t , x , k ; λ ) (43) By combining Equation (41) with the RHS of Equation (22), we get: ∞ ∑ n = 0 ∂ ∂λ { Q n ( x ; λ , k ) } t n n ! = k 2 ( 2 λ t + 1 ) ∞ ∑ n = 0 Q n ( x ; λ , k − 1 ) t n n ! + xt ∞ ∑ n = 0 Q n ( x − 1; λ , k ) t n n ! 8 Axioms 2019 , 8 , 112 which yields: ∞ ∑ n = 0 ∂ ∂λ { Q n ( x ; λ , k ) } t n n ! = k 2 ∞ ∑ n = 0 ( 2 n λ Q n − 1 ( x ; λ , k − 1 ) + Q n ( x ; λ , k − 1 )) t n n ! + x ∞ ∑ n = 0 nQ n − 1 ( x − 1; λ , k ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the following theorem: Theorem 6. Let n ∈ N . Then, we have: ∂ ∂λ { Q n ( x ; λ , k ) } = kn λ Q n − 1 ( x ; λ , k − 1 ) + k 2 Q n ( x ; λ , k − 1 ) + xnQ n − 1 ( x − 1; λ , k ) (44) By combining Equation (42) with the RHS of Equation (22), we get: ∂ ∂ t ∞ ∑ n = 0 Q n ( x ; λ , k ) t n n ! = k λ 2 2 ∞ ∑ n = 0 Q n ( x ; λ , k − 1 ) t n n ! + x λ ∞ ∑ n = 0 Q n ( x − 1; λ , k ) t n n ! which, by comparing the coefficients of t n n ! on both sides of the above equation, yields the following theorem: Theorem 7. Let n ∈ N 0 . Then, we have: Q n + 1 ( x ; λ , k ) = k λ 2 2 Q n ( x ; λ , k − 1 ) + x λ Q n ( x − 1; λ , k ) (45) By combining Equation (43) with the RHS of Equation (22) and the Taylor series of the function log ( 1 + λ t ) , we get: ∞ ∑ n = 0 ∂ ∂ x { Q n ( x ; λ , k ) } t n n ! = ∞ ∑ n = 1 ( − 1 ) n − 1 λ n t n n ∞ ∑ n = 0 Q n ( x ; λ , k ) t n n ! . Applying the Cauchy product rule to the above equation yields: ∞ ∑ n = 0 ∂ ∂ x { Q n ( x ; λ , k ) } t n n ! = t ∞ ∑ n = 0 ( n ∑ j = 0 ( − 1 ) j ] n j ) j ! λ j + 1 j + 1 Q n − j ( x ; λ , k ) ) t n n ! . Comparing the coefficients of t n n ! on both sides of the above equation, we arrive at the following theorem: Theorem 8. Let n ∈ N . Then, we have: ∂ ∂ x { Q n ( x ; λ , k ) } = n n − 1 ∑ j = 0 ( − 1 ) j ] n − 1 j ) j ! λ j + 1 j + 1 Q n − 1 − j ( x ; λ , k ) (46) Remark 1. Substituting Equation (16) into Equation (46) yields the following formula including Daehee numbers: ∂ ∂ x { Q n ( x ; λ , k ) } = n n − 1 ∑ j = 0 ] n − 1 j ) λ j + 1 D j Q n − 1 − j ( x ; λ , k ) (47) 9