Article ID: MTJPAM-D-19-00005


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-19-00005; Volume 1 / Issue 1 / Year 2019, Pages 1-76

Document Type: Research Paper

Author(s): Yilmaz Simsek a

aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya-Turkey

Received: 2 September 2019, Accepted: 9 November 2019, Available online: 28 November 2019.

Corresponding Author: Yilmaz Simsek (Email address:

Full Text: PDF


The main objective of this article is to give and classify new formulas of p-adic integrals and blend these formulas with previously well known formulas. Therefore, this article gives briefly the formulas of p-adic integrals which were found previously, as well as applying the integral equations to the generating functions and other special functions, giving proofs of the new interesting and novel formulas. The p-adic integral formulas provided in this article contain several important well-known families of special numbers and special polynomials such as the Bernoulli numbers and polynomials, the Euler numbers and polynomials, the Stirling numbers, the Lah numbers, the Peters numbers and polynomials, the central factorial numbers, the Daehee numbers and polynomials, the Changhee numbers and polynomials, the Harmonic numbers, the Fubini numbers, combinatorial numbers and sums. In addition, we defined two new sequences containing the Bernoulli numbers of the first kind and the Euler numbers of the first kind. These two sequences include central factorial numbers, Bernoulli numbers of the first kind and the Euler numbers of the first kind. Some computation formulas and identities for these sequences are given. Finally, we give further remarks, observations and comments related to content of this paper.

Keywords: p-adic q-integrals, Volkenborn integral, Generating function, Special functions, Bernoulli numbers and polynomials of the first kind, Euler numbers and polynomials of the first kind, Stirling numbers, Lah numbers, Peters numbers and polynomials, Central factorial numbers, Daehee numbers and polynomials, Changhee numbers and polynomials, Harmonic numbers, Fubini numbers, Combinatorial numbers and sums

  1. M. Acikgoz, S. Araci, On the Generating Function for Bernstein Polynomials, Amer. Institute of Physics Conference Proceedings CP1281, 1141-1144, 2010.
  2. M. Aigner, A Course in Enumeration, Springer-Verlag, Berlin-Heidelberg, 2007.
  3. Y. Amice, Mesures p-adiques, Séminaire Delange-Pisot-Poitou, Théorie des nombres 6 (2), 1-6, 1964-1965.
  4. Y. Amice, Integration p-adique Selon A. Volkenborn, Séminaire Delange-Pisot-Poitou, Théorie des nombres 13 (2), G4, G1-G9, 1971-1972.
  5. S. Araci, M. Acikgoz, E. Sen, On the extended Kim’s p-adic q-deformed fermionic integrals in the p-adic integer ring, J. Number Theory 133 (10), 3348-3361, 2013.
  6. S. Araci, U. Duran, M. Acikgoz, (p, q)-Volkenborn integration, J. Number Theory 171, 18-30, 2017.
  7. T. M. Apostol, On the Lerch Zeta Function, Pacific J. Math. 1 (2), 161-167, 1951.
  8. A. Bayad, Y. Simsek, H. M. Srivastava, Some Array Type Polynomials Associated with Special Numbers and Polynomials, Appl. Math. Compute. 244, 149-157, 2014.
  9. H. Belbachir, I. E. Bousbaa, Associated Lah Numbers and r-Stirling Numbers, arXiv:1404.5573v2 (math.CO), 12 May 2014.
  10. K. N. Boyadzhiev, Close Encounters with the Stirling Numbers of the Second Kind, Math. Mag. 85, 252-266, 2012.
  11. K. N. Boyadzhiev, Binomial Transform and the Backward Difference,
  12. L. Carlitz, The Reciprocity Theorem for Dedekind Sums, Pacific J. Math. 3, 523-527, 1953.
  13. C. H. Chang, C. W. Ha, A Multiplication Theorem for the Lerch Zeta Function and Explicit Representations of the Bernoulli and Euler Polynomials, J. Math. Anal. Appl. 315, 758-767, 2006.
  14. C. A. Charalambides, Combinatorial Methods in Discrete Distributions, A John Wiley and Sons, Inc., Publication, 2015.
  15. C. A. Charalambides, Enumerative Combinatorics, Chapman and Hall/Crc, Press Company, London, New York, 2002.
  16. J. Cigler, Fibonacci Polynomials and Central Factorial Numbers, preprint.
  17. J. Choi, H.M. Srivastava, Some Summation Formulas Involving Harmonic Numbers and Generalized Harmonic Numbers, Mathematical and Computer Modelling 54, 2220-2234, 2011.
  18. L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Reidel, Dordrecht and Boston, 1974.
  19. H. Coskun, Multiple Bracket Function, Stirling Number, and Lah Number Identities, arXiv:1212.6573v2 (math.NT) 17 Jun 2015.
  20. G. E. W. Dickson, L.E., H.H. Mitchell, H.S. Vandiver, Algebraic Numbers. Report of the Committee on Algebraic Numbers, National Research Council, Bulletin of the National Research Council Vol. 5, Part 3. Number 28, National Academy of Sciences, Washington, 1923.
  21. G. B. Djordjevic, G. V. Milovanovic, Special Classes of Polynomials, University of Nis, Faculty of Technology Leskovac, 2014.
  22. K. N. Boyadzhiev, Lah numbers, Laguerre Polynomials of Order Negative One, and the n th Derivative of  exp(\frac{1}{x}), Acta Univ. Sapientiae, Mathematica 8 (1), 22-31, 2016.
  23. R. A. Brigham II, A Harmonic M-Factorial Function and Applications, Doctoral Dissertations, 2557, 2017,
  24. P. L. Butzer, K. Schmidt, E. L. Stark, L. Vogt, Central Factorial Numbers; Their Main Properties and Some Applications, Numer. Funct. Anal. and Optimiz. 10 (5-6), 419-488, 1989.
  25. P. F. Byrd, New Relations Between Fibonacci and Bernoulli Numbers, Fibonacci Quarterly 13, 111-114, 1975.
  26. N. P. Cakic, G. V. Milovanovic, On Generalized Stirling Numbers and Polynomials, Mathematica Balkanica 18, 241-248, 2004.
  27. Y. Do, D. Lim, On (h, q)-Daehee Numbers and Polynomials, Adv. Difference Equ. 107, 1-9, 2015.
  28. B. S. El-Desouky, A. Mustafa, New Results and Matrix Representation for Daehee and Bernoulli Numbers and Polynomials, Applied Mathematical Sciences 9 (73), 3593-3610, 2015, arXiv:1412.8259v1 (math.CO) 29 Dec 2014.
  29. B. S. El-Desouky, R. S. Goma, Multiparameter Poly-Cauchy and Poly-Bernoulli Numbers and Polynomials, International J. Mathematical Analysis 9 (53), 2619-2633, 2015, arXiv:1410.5300v1 (math.CO) 20 Oct 2014.
  30. A. Garsia, J. Remmel, A Combinatorial Interpretation of q-Derangement and q-Laguerre Numbers, European J. Combin. 1, 47-59, 1980.
  31. R. Golombek, Aufgabe 1088, El. Math. 49, 126-127, 1994.
  32. I. J. Good, The Number of Ordering of n Candidates When Ties are Permitted, Fibonacci Quart. 13, 11-18, 1975.
  33. H. W. Gould, Fundamentals of Series,
  34. H. W. Gould, Combinatorial Numbers and Associated Identities,
  35. H.W. Gould, Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations, Revised ed., Morgantown Printing and Binding Company, Morgantown, West Virginia, 1972.
  36. B. N. Guo, F. Qi, An Explicit Formula for Bernoulli Numbers in Terms of Stirling Numbers of the Second Kind, J. Ana. Num. Theor. 3 (1), 27-30, 2015.
  37. K. Iwasawa, Lectures on p-adic L-function, Princeton Univ. Press, Princeton, 1972.
  38. L.-C. Jang, W. Kim, H.-I. Kwon, On degenerate Daehee polynomials and numbers of the third kind, J. Computational Appl. Math., 2019,
  39. L. C. Jang, T. Kim, A New Approach to q-Euler Numbers and Polynomials, J. Concr. Appl. Math. 6, 159-168, 2008.
  40. L. C. Jang, T. Kim, D. H. Lee, D. W. Park, An Application of Polylogarithms in the Analogs of Genocchi Numbers, Notes Number Theory Discrete Math. 7 (3), 65-69, 2001.
  41. L. C. Jang, H. K. Pak, Non-archimedean Integration Associated with q-Bernoulli Numbers, Proc. Jangjeon Math. Soc. 5 (2), 125-129, 2002.
  42. H. Jolany, H. Sharifi, R. E. Alikelaye, Some Results for the Apostol-Genocchi Polynomials of Higher Order, Bull. Malays. Math. Sci. Soc. 36 (2), 465-479, 2013.
  43. J. Jeong, D.-J. Kang, S.-H. Rim, Symmetry Identities of Changhee Polynomials of Type Two, Symmetry 10, 740, 2018.
  44. J.-W. Park, G.-W. Jang, J. Kwon, The λ-analogue degenerate Changhee polynomials and numbers, Global J. Pure Appl. Math. ISSN 0973-1768 13 (3), 893-900, 2017.
  45. C. Jordan, Calculus of Finite Differences (2nd ed.), Chelsea Publishing Company, New York, 1950.
  46. N. Kilar, Y. Simsek, A New Family of Fubini Type Numbers and Polynomials Associated with Apostol-Bernoulli Numbers and Polynomials, J. Korean Math. Soc. 54 (5), 1605-1621, 2017.
  47. D. S. Kim, D. V. Dolgy, D. Kim, T. Kim, Some Identities on r-central Factorial Numbers and r-central Bell Polynomials, Adv. Difference Equ. 245, 1-11, 2019.
  48. D. S. Kim, T. Kim, Some New Identities of Frobenius-Euler Numbers and Polynomials, J. Ineq. Appl. 307, 1-10, 2012.
  49. D. S. Kim, T. Kim, Daehee Numbers and Polynomials, Appl. Math. Sci. (Ruse) 7 (120), 5969-5976, 2013.
  50. D. S. Kim, T. Kim, A Note on Boole Polynomials, Integral Transforms Spec. Funct. 25 (8), 627-633, 2014.
  51. D. S. Kim, T. Kim, Some Identities of Degenerate Special Polynomials, Open Math. 13, 380-389, 2015.
  52. D. S. Kim, T. Kim, J. Seo, A note on Changhee numbers and polynomials, Adv. Stud. Theor. Phys. 7, 993-1003, 2013.
  53. D. S. Kim, T. Kim, J. J. Seo, T. Komatsu, Barnes’ Multiple Frobenius-Euler and Poly-Bernoulli Mixed-type Polynomials, Adv. Difference Equ. 92, 1-16, 2014.
  54. T. Kim, An Analogue of Bernoulli Numbers and Their Congruences, Rep. Fac. Sci. Engrg. Saga Univ. Math. 22 (2), 21-26, 1994.
  55. T. Kim, The Modified q-Euler Numbers and Polynomials, arXiv:math/0702523v1 (math.NT) 18 Feb 2007.
  56. T. Kim, On a q-analogue of the p-adic log gamma functions, J. Number Theory 76, 320-329, 1999.
  57. T. Kim, q-Volkenborn Integration, Russ. J. Math. Phys. 19, 288-299, 2002.
  58. T. Kim, An Invariant p-adic Integral Associated with Daehee Numbers, Integral Transforms Spec. Funct. 13 (1), 65-69, 2002.
  59. T. Kim, q-Euler Numbers and Polynomials Associated with p-adic q-Integral and Basic q-zeta Function, Trend Math. Information Center Math. Sciences 9, 7-12, 2006.
  60. T. Kim, On the Analogs of Euler Numbers and Polynomials Associated with p-adic q-integral on p at q = 1, J. Math. Anal. Appl. 331 (2), 779-792, 2007.
  61. T. Kim, An Invariant p-adic q-integral on p, Appl. Math. Letters 21 , 105-108, 2008.
  62. T. Kim, p-adic l-functions and Sums of Powers, arXiv:math/0605703v1 (math.NT) 27 May 2006.
  63. T. Kim, On the q-extension of Euler and Genocchi Numbers, J. Math. Anal. Appl. 326 (2), 1458-1465, 2007.
  64. T. Kim, A Note on Central Factorial Numbers, Proceed. Jangjeon Math. Soc. 21 (4), 575-588, 2018.
  65. D. Kim, H. O.Ayna, Y. Simsek, A. Yardimci, New Families of Special Numbers and Polynomials Arising from Applications of p-adic q-integrals, Adv. Difference Equ. 207, 1-11, 2017.
  66. T. Kim, J. Choi, Y. H. Kim, C. S. Ryoo, On the Fermionic p-adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials, J. Inequal. Appl. Article ID 864247, 1-12, 2010.
  67. T. Kim, D. S. Kim, D. V. Dolgy, J. J. Seo, Bernoulli Polynomials of the Second Kind and Their Identities Arising from Umbral Calculus, J. Nonlinear Sci. Appl. 9, 860-869, 2016.
  68. T. Kim, D. S. Kim, K. W. Hwang, Some Identities of Laguerre Polynomials Arising from Differential Equations, Adv. Differ. Equ. 159, 1-9, 2016.
  69. T. Kim, D. S. Kim, G-W. Jang, J. Kwon, Symmetric Identities for Fubini Polynomials, Symmetry 10 (6), 219, 2018,
  70. T. Kim, M.S. Kim, L.C. Jang, S.-H. Rim, New q-Euler Numbers and Polynomials Associated with p-adic q-integrals,
  71. T. Kim, S. H. Rim, Some q-Bernoulli Numbers of Higher Order Associated with the p-adic q-integrals, Indian J. Pure Appl. Math. 32 (10), 1565-1570, 2001.
  72. T. Kim, S.H. Rim, Y. Simsek, D. Kim, On the Analogs of Bernoulli and Euler Numbers, Related Identities and Zeta and l-functions, J. Korean Math. Soc. 45 (2), 435-453, 2008.
  73. M. S. Kim, On Euler Numbers, Polynomials and Related p-adic Integrals, J. Number Theory 129 (9), 2166-2179, 2009.
  74. M. S. Kim, J. W. Son, Analytic Properties of the q-Volkenborn Integral on the Ring of p-adic Integers, Bull. Korean Math. Soc. 44 (1), 1-12, 2007.
  75. M. S. Kim, J. W. Son, Some Remarks on a q-analogue of Bernoulli Numbers, J. Korean Math.Soc. 39 (2), 221-236, 2002.
  76. A. Khrennikov, p-adic Valued Distributions and Their Applications to the Mathematical Physics, Kluwer, Dordreht, 1994.
  77. I. Kleiner, A History of Abstract Algebra, Birkhauser, Boston, 2007.
  78. N. Koblitz, p-Adic Numbers, p-adic Analysis, and Zeta-Functions (Second Edition), Springer-Verlag, New Yook, Beriln, Haidellerg, 1977.
  79. T. Kubota, H.W. Leopoldt, Eine p-adische theorie der zetawerte I, J. Reine Angew. Math. 214-215, 328-339, 1964.
  80. S. Lang, Cyclotomic Fields, Springer Verlag, New York, 1978.
  81. D. Lim, On the Twisted Modified q-Daehee Numbers and Polynomials, Adv. Stud. Theor. Phys. 9 (4), 199-211, 2015.
  82. G. G. Lorentz, Bernstein Polynomials, Chelsea Publishing Company, New York, 1986.
  83. Q. M. Luo, H. M. Srivastava, Some Generalizations of the Apostol-Genocchi Polynomials and the Stirling Numbers of the Second Kind, Appl. Math. Compute. 217, 5702-5728, 2011.
  84. D. Merlini, R. Sprugnoli, M. C. Verri, The Cauchy Numbers, Discrete Math. 306 (16), 1906-1920, 2006.
  85. B. Osgood, W. Wu, Falling Factorials, Generating Functions, and Conjoint Ranking Tables, J. Integer Seq. 12, 1-13, 2009.
  86. H. Ozden, p-adic q-measure and its applications, Doctoral dissertation, Uludag University, Bursa, Turkey, 2009.
  87. H. Ozden, Y. Simsek, Modification and Unification of the Apostol-type Numbers and Polynomials and Their Applications, Appl. Math. Compute. 235, 338-351, 2009.
  88. H. Ozden, Y. Simsek, I. N. Cangul, Euler Polynomials Associated with p-adic q-Euler Measure, Gen. Math. 15 (2-3), 2007.
  89. J. W. Park, On a q-analogue of (h, q)-Daehee Numbers and Polynomials of Higher Order, J. Compute. Analy. Appl. 21 (1), 769-776, 2016.
  90. A. P. Prudnikov, Yu. A. Bryckov, O. I. Maricev, Integrals and Series, Vol. 1: Elementary Functions, Nauka, Moscow, 1981, (in Russian); Translated from the Russian and with a Preface by N.M. Queen, Gordon and Breach Science Publishers, New York, Philadelphia, London, Paris, Montreux, Tokyo and Melbourne, 1986.
  91. F. Qi, Explicit Formulas for Computing Bernoulli Numbers of the Second Kind and Stirling Numbers of the First Kind, Filomat 28 (2), 319-327, 2014.
  92. F. Qi, X. T. Shi, F. F. Liu, Several Identities Involving the Falling and Rising Factorials and the Cauchy, Lah, and Stirling Numbers, Acta Univ. Sapientiae, Mathematica 8 (2), 282-297, 2016.
  93. E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
  94. K. F. Riley, M. P. Hobson, S. J. Bence, Mathematical Methods for Physics and Engineering: A Comprehensive Guide (Third Edition), Cambridge University Press, New York, 2006.
  95. S.-H. Rim, T. Kim, A Note on p-adic Euler Measure on Zp, Russ. J. Math. Phys. 13 (3), 2006.
  96. S.-H. Rim, T. Kim, S.S. Pyo, Identities between harmonic, hyperharmonic and Daehee numbers, J. Inequal Appl. 1, 168, 2018.
  97. J. Riordan, Introduction to Combinatorial Analysis, Princeton University Press, 1958.
  98. A. M. Robert, A Course in p-adic Analysis, Springer, New York, 2000.
  99. S. Roman, The Umbral Calculus, Dover Publ. Inc., New York, 2005.
  100. C. S. Ryoo, D. V. Dolgy, H. I. Kwon, Y. S. Jang, Functional Equations Associated with Generalized Bernoulli Numbers and Polynomials, Kyungpook Math. J. 55, 29-39, 2015.
  101. W. H. Schikhof, Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge Studies in Advanced Mathematics 4, Cambridge University Press, Cambridge, 1984.
  102. K. Shiratani, S. Yokoyama, An Application of p-adic Convolutions, Mem. Fac. Sci. Kyushu Univ. Ser. A Math. 36 (1), 73-83, 1982.
  103. Y. Simsek, Twisted p-adic (h, q)L-functions, Comput. Math. Appl. 59 (6), 2097-2110, 2010.
  104. Y. Simsek, Generating Functions for Generalized Stirling Type Numbers, Array Type Polynomials, Eulerian Type Polynomials and Their Applications, Fixed Point Theory Appl. 87, 1-28, 2013.
  105. Y. Simsek, Functional Equations from Generating Functions: A Novel Approach to Deriving Identities for the Bernstein Basis Functions, Fixed Point Theory Appl. 80, 1-13, 2013.
  106. Y. Simsek, Identities Associated with Generalized Stirling Type Numbers and Eulerian Polynomials, Math. Comput. Appl. 18 (3), 251-263, 2013.
  107. Y. Simsek, Special Numbers on Analytic Functions, Applied Math. 5, 1091-1098, 2014.
  108. Y. Simsek, Analysis of the Bernstein Basis Functions: An Approach to Combinatorial Sums Involving Binomial Coefficients and Catalan Numbers, Math. Method. Appl. Sci. 38, 3007-3021, 2015.
  109. Y. Simsek, Computation Methods for Combinatorial Sums and Euler Type Numbers Related to New Families of Numbers, Math. Meth. Appl. Sci. 40 (7), 2347-2361, 2016.
  110. Y. Simsek, Apostol Type Daehee Numbers and Polynomials, Adv. Studies Contemp. Math. 26 (3), 1-12, 2016.
  111. Y. Simsek, Analysis of the p-adic q-Volkenborn Integrals: An Approach to Generalized Apostol-type Special Numbers and Polynomials and Their Applications, Cogent Math. 3, 1-17, 2016.
  112. Y. Simsek, Identities on the Changhee Numbers and Apostol-Daehee Polynomials, Adv. Stud. Contemp. Math. 27 (2), 199-212, 2017.
  113. Y. Simsek, Formulas for p-adic q-integrals Including Falling-Rising Factorials, Combinatorial Sums and Special Numbers, arXiv:1702.06999v1 (math.NT) 22 Feb 2017.
  114. Y. Simsek, New families of Special Numbers for Computing Negative Order Euler Numbers and Related Numbers and Polynomials, Appl. Anal. Discrete Math. 12, 1-35, 2018.
  115. Y. Simsek, Construction of Some New Families of Apostol-type Numbers and Polynomials via Dirichlet Character and p-adic q-integrals, Turk. J. Math. 42, 557-577, 2018.
  116. Y. Simsek, Construction method for generating functions of special numbers and polynomials arising from analysis of new operators, Math. Meth. Appl. Sci. 41, 6934-6954, 2018.
  117. Y. Simsek, Peters Type Polynomials and Numbers and Their Generating Functions: Approach with p-adic Integral Method, Math Meth Appl Sci., 1-17, 2019, doi:10.1002/mma.5807.
  118. Y. Simsek, A New Family of Combinatorial Numbers and Polynomials Associated with Peters Numbers and Polynomials, to appear in Appl. Anal. Discrete Math.
  119. Y. Simsek , M. Acikgoz , A New Generating Function of (q-) Bernstein-type Polynomials and Their Interpolation Function, Abstr. Appl. Anal. 769095, 1-12, 2010.
  120. Y. Simsek, H. M. Srivastava, A Family of p-adic Twisted Interpolation Functions Associated with the Modified Bernoulli Numbers, Appl. Math. Comput. 216 (10), 2976-2987, 2010.
  121. Y. Simsek, S.H. Rim, L.-C. Jang, D. J. Kang, J.J. Seo, A Note on q-Daehee Sums, Proc. 16th Int. Conf. Jangjeon Math. Soc. 16, 159-166, 2005.
  122. H. M. Srivastava, Some Generalizations and Basic or q-extensions of the Bernoulli, Euler and Genocchi Polynomials, Appl. Math. Inf. Sci. 5, 390-444, 2011.
  123. H. M. Srivastava, J. Choi, Zeta and q-zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  124. H. M. Srivastava, T. Kim, Y. Simsek, q-Bernoulli Numbers and Polynomials Associated with Multiple q-zeta Functions and Basic L-series, Russ. J. Math. Phys. 12, 241-268, 2005.
  125. H. M. Srivastava, G. D. Liu, Some Identities and Congruences Involving a Certain Family of Numbers, Russ. J. Math. Phys. 16, 536-542, 2009.
  126. E. Steinitz, Algebraische Theorie der Körper (2nd ed.), Chelsea Publishing Company, Chelsea, 1950.
  127. B. A. Tangedal, P. T.Young, On p-adic multiple zeta and log gamma functions, J. Number Theory 131 (7), 1240-1257, 2011.
  128. N. M. Temme, Asymptotic Estimates for Laguerre Polynomials, J. Appl. Math. Physics 41, 114-126, 1990.
  129. V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.
  130. A. Volkenborn, On Generalized p-adic Integration, Mém. Soc. Math. Fr. 39-40, 375-384, 1974.
  131. C.F. Woodcock, Convolutions on the Ring of p-adic Integers, J. Lond. Math. Soc. 20 (2), 101-108, 1979.
  132. H. Wang, G. Liu, An Explicit Formula for Higher Order Bernoulli Polynomials of the Second Kind, Integers 13, #A75, 2013.
  133. G. E. Wahlin, A New Development of the Theory of Algebraic Numbers, Transactions of the American Mathematical Society 16 (4), 502-508, 1915.
  134. S. J. Yun, J.W. Park, On the fully degenerate Daehee numbers and polynomials of the second kind, Preprints ( 8 October 2018, doi:10.20944/preprints201810.0129.v1.