Euler matrices and their algebraic properties revisited
Date
2020
2020
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Abstract
This paper addresses the generalized Euler polynomial matrix E (α) (x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α) (x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices.
Referencias:
[1] T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, London UK, 51-63, (2014).
[2] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993).
[3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl., 329, 49-59 (2001).
[4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences, Linear Algebra Appl., 357, 247-258 (2002).
[5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Boston US, 204-219, (1974).
[6] T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh¨auser, London UK, 97-167, (2012).
[7] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272, 251-257 (2014).
[8] Y. He, S. Araci and H. M. Srivastava, Some new formulas for the products of the Apostol type polynomials, Adv. Differ. Equ., 2016, 1-18 (2016).
[9] Y. He, S. Araci, H. M. Srivastava and M. Acikg¨oz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Appl. Math. Comput., 262, 31-41 (2015).
[10] P. Hern´andez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68, 203-225 (2015).
[11] G. I. Infante, J. L. Ram´ırez and A. S¸ahin, Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices, Math. Rep. (Bucur.), 19(69), 399-417 (2017).
[12] D. S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. Appl., 2014:1, 1-12 (2014).
[13] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart., 40, 203-211 (2002).
[14] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130, 527-534 (2003).
[15] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302 (2005).
[16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin DE, 120-162, (1924).
[17] A. Pint´er and H. M. Srivastava, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math., 85, 483-495 (2013).
[18] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo 55, 1-29 (2018).
[19] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York US, 99-127, (1968).
[20] H. M. Srivastava, M. A. Boutiche and M. Rahmani, A class of Frobenius-type Eulerian polynomials, Rocky Mountain J. Math., 48, 1003-1013 (2018).
[21] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London UK, 76-140, (2012).
[22] H. M. Srivastava, I. Kucuko˘glu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory, 181, 117-146 (2017).
[23] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex UK, 384-402, (1984).
[24] H. M. Srivastava, M. Masjed-Jamei and M. Reza Beyki, A parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Appl. Math. Inform. Sci., 12, 907-916 (2018).
[25] H. M. Srivastava, M. A. Ozarslan and C. Kaanuglu, Some ¨ generalized Lagrange-based Apostol-Bernoulli, ApostolEuler and Apostol-Genocchi polynomials, Russian J. Math. Phys., 20, 110-120 (2013).
[26] H. M. Srivastava, M. A. Ozarslan and B. Yilmaz, Some ¨families of differential equations associated with the Hermitebased Appell polynomials and other classes of Hermite-based polynomials, Filomat, 28, 695-708 (2014).
[27] H. M. Srivastava and A. Pint´er, Remarks on some ´relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17, 375-380 (2004).
[28] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math., 156, 2606-2619 (2008).
[29] Z. Z. Zhang, The linear algebra of generalized Pascal matrix, Linear Algebra Appl., 250, 51-60 (1997).
[30] Y. Yang and C. Micek, Generalized Pascal function matrix and its applications, Linear Algebra Appl., 423, 230-245 (2007).
[31] Z. Z. Zhang and M. X. Liu, An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271, 169-177 (1998).
[32] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math., 154, 1622-1632 (2006).
[33] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math., 38, 457- 465 (2007).
[34] D. Zheng, q-analogue of the Pascal matrix, Ars Combin., 87, 321-336 (2008).
[1] T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, London UK, 51-63, (2014).
[2] G. S. Call and D. J. Velleman, Pascal’s matrices, Amer. Math. Monthly, 100, 372-376 (1993).
[3] G.-S. Cheon and J.-S. Kim, Stirling matrix via Pascal matrix, Linear Algebra Appl., 329, 49-59 (2001).
[4] G.-S. Cheon and J.-S. Kim, Factorial Stirling matrix and related combinatorial sequences, Linear Algebra Appl., 357, 247-258 (2002).
[5] L. Comtet, Advanced Combinatorics. The Art of Finite and Infinite Expansions, D. Reidel Publishing Co., Boston US, 204-219, (1974).
[6] T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh¨auser, London UK, 97-167, (2012).
[7] B.-N. Guo and F. Qi, Explicit formulae for computing Euler polynomials in terms of Stirling numbers of the second kind, J. Comput. Appl. Math., 272, 251-257 (2014).
[8] Y. He, S. Araci and H. M. Srivastava, Some new formulas for the products of the Apostol type polynomials, Adv. Differ. Equ., 2016, 1-18 (2016).
[9] Y. He, S. Araci, H. M. Srivastava and M. Acikg¨oz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials. Appl. Math. Comput., 262, 31-41 (2015).
[10] P. Hern´andez-Llanos, Y. Quintana and A. Urieles, About extensions of generalized Apostol-type polynomials, Results Math., 68, 203-225 (2015).
[11] G. I. Infante, J. L. Ram´ırez and A. S¸ahin, Some results on q-analogue of the Bernoulli, Euler and Fibonacci matrices, Math. Rep. (Bucur.), 19(69), 399-417 (2017).
[12] D. S. Kim and T. Kim, Some identities of q-Euler polynomials arising from q-umbral calculus, J. Inequal. Appl., 2014:1, 1-12 (2014).
[13] G.-Y. Lee, J.-S. Kim and S.-G. Lee, Factorizations and eigenvalues of Fibonacci and symmetric Fibonacci matrices, Fibonacci Quart., 40, 203-211 (2002).
[14] G.-Y. Lee, J.-S. Kim and S.-H. Cho, Some combinatorial identities via Fibonacci numbers, Discrete Appl. Math., 130, 527-534 (2003).
[15] Q.-M. Luo and H. M. Srivastava, Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials, J. Math. Anal. Appl., 308, 290-302 (2005).
[16] N. E. Nørlund, Vorlesungen ¨uber Differenzenrechnung, Springer-Verlag, Berlin DE, 120-162, (1924).
[17] A. Pint´er and H. M. Srivastava, Addition theorems for the Appell polynomials and the associated classes of polynomial expansions, Aequationes Math., 85, 483-495 (2013).
[18] Y. Quintana, W. Ram´ırez and A. Urieles, On an operational matrix method based on generalized Bernoulli polynomials of level m, Calcolo 55, 1-29 (2018).
[19] J. Riordan, Combinatorial Identities, John Wiley & Sons, Inc., New York US, 99-127, (1968).
[20] H. M. Srivastava, M. A. Boutiche and M. Rahmani, A class of Frobenius-type Eulerian polynomials, Rocky Mountain J. Math., 48, 1003-1013 (2018).
[21] H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier, London UK, 76-140, (2012).
[22] H. M. Srivastava, I. Kucuko˘glu and Y. Simsek, Partial differential equations for a new family of numbers and polynomials unifying the Apostol-type numbers and the Apostol-type polynomials, J. Number Theory, 181, 117-146 (2017).
[23] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Horwood Ltd., West Sussex UK, 384-402, (1984).
[24] H. M. Srivastava, M. Masjed-Jamei and M. Reza Beyki, A parametric type of the Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials, Appl. Math. Inform. Sci., 12, 907-916 (2018).
[25] H. M. Srivastava, M. A. Ozarslan and C. Kaanuglu, Some ¨ generalized Lagrange-based Apostol-Bernoulli, ApostolEuler and Apostol-Genocchi polynomials, Russian J. Math. Phys., 20, 110-120 (2013).
[26] H. M. Srivastava, M. A. Ozarslan and B. Yilmaz, Some ¨families of differential equations associated with the Hermitebased Appell polynomials and other classes of Hermite-based polynomials, Filomat, 28, 695-708 (2014).
[27] H. M. Srivastava and A. Pint´er, Remarks on some ´relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett., 17, 375-380 (2004).
[28] P. Stanimirovi´c, J. Nikolov and I. Stanimirovi´c, A generalization of Fibonacci and Lucas matrices, Discrete Appl. Math., 156, 2606-2619 (2008).
[29] Z. Z. Zhang, The linear algebra of generalized Pascal matrix, Linear Algebra Appl., 250, 51-60 (1997).
[30] Y. Yang and C. Micek, Generalized Pascal function matrix and its applications, Linear Algebra Appl., 423, 230-245 (2007).
[31] Z. Z. Zhang and M. X. Liu, An extension of generalized Pascal matrix and its algebraic properties, Linear Algebra Appl., 271, 169-177 (1998).
[32] Z. Zhang and J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math., 154, 1622-1632 (2006).
[33] Z. Zhang and Y. Zhang, The Lucas matrix and some combinatorial identities, Indian J. Pure Appl. Math., 38, 457- 465 (2007).
[34] D. Zheng, q-analogue of the Pascal matrix, Ars Combin., 87, 321-336 (2008).
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