## Generalized Riordan arrays.(English)Zbl 1158.05008

Summary: We generalize the concept of Riordan array. A generalized Riordan array with respect to $$c_n$$ is an infinite, lower triangular array determined by the pair $$(g(t),f(t))$$ and has the generic element $$d_{n,k}=[t^n/c_n]g(t)(f(t))^k/c_k$$, where $$c_n$$ is a fixed sequence of non-zero constants with $$c_{0}=1$$.
We demonstrate that the generalized Riordan arrays have similar properties to those of the classical Riordan arrays. Based on the definition, the iteration matrices related to the Bell polynomials are special cases of the generalized Riordan arrays and the set of iteration matrices is a subgroup of the Riordan group. We also study the relationships between the generalized Riordan arrays and the Sheffer sequences and show that the Riordan group and the group of Sheffer sequences are isomorphic. From the Sheffer sequences, many special Riordan arrays are obtained. Additionally, we investigate the recurrence relations satisfied by the elements of the Riordan arrays. Based on one of the recurrences, some matrix factorizations satisfied by the Riordan arrays are presented. Finally, we give two applications of the Riordan arrays, including the inverse relations problem and the connection constants problem.

### MSC:

 05A15 Exact enumeration problems, generating functions 05A40 Umbral calculus 05A19 Combinatorial identities, bijective combinatorics
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### References:

 [1] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press Cambridge [2] Bayat, M.; Teimoori, H., The linear algebra of the generalized Pascal functional matrix, Linear algebra appl., 295, 1-3, 81-89, (1999) · Zbl 0935.15022 [3] Boas, R.P.; Buck, R.C., Polynomials defined by generating relations, Amer. math. monthly, 63, 626-632, (1956) · Zbl 0073.05802 [4] Brawer, R.; Pirovino, M., The linear algebra of the Pascal matrix, Linear algebra appl., 174, 13-23, (1992) · Zbl 0755.15012 [5] Brown, J.W., Generalized Appell connection sequences. II, J. math. anal. appl., 50, 458-464, (1975) · Zbl 0301.33009 [6] Brown, J.W.; Goldberg, J.L., Generalized Appell connection sequences, J. math. anal. appl., 46, 242-248, (1974) · Zbl 0276.33027 [7] Brown, J.W.; Kuczma, M., Self-inverse Sheffer sequences, SIAM J. math. anal., 7, 5, 723-728, (1976) · Zbl 0335.33017 [8] Cheon, G.-S., A note on the Bernoulli and Euler polynomials, Appl. math. lett., 16, 3, 365-368, (2003) · Zbl 1055.11016 [9] Cheon, G.-S.; Hwang, S.-G.; Lee, S.-G., Several polynomials associated with the harmonic numbers, Discrete appl. math., 155, 18, 2573-2584, (2007) · Zbl 1130.11011 [10] Cheon, G.-S.; Kim, J.-S., Stirling matrix via Pascal matrix, Linear algebra appl., 329, 1-3, 49-59, (2001) · Zbl 0988.05009 [11] Cheon, G.-S.; Kim, J.-S., Factorial Stirling matrix and related combinatorial sequences, Linear algebra appl., 357, 247-258, (2002) · Zbl 1016.05004 [12] Comtet, L., Advanced combinatorics, (1974), D. Reidel Publishing Co. Dordrecht [13] Corsani, C.; Merlini, D.; Sprugnoli, R., Left-inversion of combinatorial sums, Discrete math., 180, 1-3, 107-122, (1998) · Zbl 0903.05005 [14] Egorychev, G.P.; Zima, E.V., Decomposition and group theoretic characterization of pairs of inverse relations of the Riordan type, Acta appl. math., 85, 1-3, 93-109, (2005) · Zbl 1074.05012 [15] Hsu, L.C., Generalized Stirling number pairs associated with inverse relations, Fibonacci quart., 25, 4, 346-351, (1987) · Zbl 0632.10011 [16] Kettle, S.G., Families enumerated by the schröder – etherington sequence and a renewal array it generates, (), 244-274 · Zbl 0523.05027 [17] Ma, X., Inverse chains of the Riordan group and their applications to combinatorial sums, J. math. res. exposition, 19, 2, 445-451, (1999) · Zbl 0940.05003 [18] D. Merlini, R. Sprugnoli, M.C. Verri, Human and constructive proof of combinatorial identities: an example from Romik, in: 2005 International Conference on Analysis of Algorithms, pp. 383-391 · Zbl 1100.05003 [19] Merlini, D.; Sprugnoli, R.; Verri, M.C., The akiyama – tanigawa transformation, Integers, 5, 1, (2005), A5 12 pp · Zbl 1087.11012 [20] Merlini, D.; Sprugnoli, R.; Verri, M.C., Lagrange inversion: when and how, Acta appl. math., 94, 3, 233-249, (2006) · Zbl 1108.05008 [21] Merlini, D.; Sprugnoli, R.; Verri, M.C., The Cauchy numbers, Discrete math., 306, 16, 1906-1920, (2006) · Zbl 1098.05008 [22] Merlini, D.; Sprugnoli, R.; Verri, M.C., The method of coefficients, Amer. math. monthly, 114, 1, 40-57, (2007) · Zbl 1191.05006 [23] Niven, I., Formal power series, Amer. math. monthly, 76, 871-889, (1969) · Zbl 0184.29603 [24] Peart, P.; Woodson, L., Triple factorization of some Riordan matrices, Fibonacci quart., 31, 2, 121-128, (1993) · Zbl 0778.05005 [25] Riordan, J., Combinatorial identities, reprint of the 1968 original, (1979), Robert E. Krieger Publishing Co. Huntington, NY [26] Rogers, D.G., Pascal triangles, Catalan numbers and renewal arrays, Discrete math., 22, 3, 301-310, (1978) · Zbl 0398.05007 [27] Roman, S., The theory of the umbral calculus. I, J. math. anal. appl., 87, 1, 58-115, (1982) · Zbl 0499.05009 [28] Roman, S., The umbral calculus, (1984), Academic Press, Inc. New York · Zbl 0536.33001 [29] Roman, S.; Rota, G.-C., The umbral calculus, Adv. math., 27, 2, 95-188, (1978) · Zbl 0375.05007 [30] Rota, G.-C.; Kahaner, D.; Odlyzko, A., On the foundations of combinatorial theory. VIII. finite operator calculus, J. math. anal. appl., 42, 684-760, (1973) · Zbl 0267.05004 [31] Shapiro, L.W., Bijections and the Riordan group, Theoret. comput. sci., 307, 2, 403-413, (2003) · Zbl 1048.05008 [32] Shapiro, L.W.; Getu, S.; Woan, W.J.; Woodson, L.C., The Riordan group, Discrete appl. math., 34, 1-3, 229-239, (1991) · Zbl 0754.05010 [33] Sheffer, I.M., Some properties of polynomial sets of type zero, Duke math. J., 5, 590-622, (1939) · Zbl 0022.01502 [34] Sheffer, I.M., Note on Appell polynomials, Bull. amer. math. soc., 51, 739-744, (1945) · Zbl 0060.19212 [35] Sprugnoli, R., Riordan arrays and combinatorial sums, Discrete math., 132, 1-3, 267-290, (1994) · Zbl 0814.05003 [36] Sprugnoli, R., Riordan arrays and the abel – gould identity, Discrete math., 142, 1-3, 213-233, (1995) · Zbl 0832.05007 [37] Srivastava, H.M.; Pintér, Á., Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. math. lett., 17, 4, 375-380, (2004) · Zbl 1070.33012 [38] Tan, M.; Wang, T., Lah matrix and its algebraic properties, Ars combin., 70, 97-108, (2004) · Zbl 1092.05500 [39] Wang, W.; Wang, T., Matrices related to the Bell polynomials, Linear algebra appl., 422, 1, 139-154, (2007) · Zbl 1112.05010 [40] W. Wang, T. Wang, A note on the relationships between the generalized Bernoulli and Euler polynomials, Ars Combin. (in press) · Zbl 1224.11035 [41] W. Wang, T. Wang, Matrices related to the idempotent numbers and the numbers of planted forests, Ars Combin. (in press) · Zbl 1249.05260 [42] Zhang, Z., The linear algebra of the generalized Pascal matrix, Linear algebra appl., 250, 51-60, (1997) · Zbl 0873.15014 [43] Zhang, Z.; Liu, M., An extension of the generalized Pascal matrix and its algebraic properties, Linear algebra appl., 271, 169-177, (1998) · Zbl 0892.15018 [44] Zhao, X.; Ding, S.; Wang, T., Some summation rules related to the Riordan arrays, Discrete math., 281, 1-3, 295-307, (2004) · Zbl 1042.05009 [45] Zhao, X.; Wang, T., The algebraic properties of the generalized Pascal functional matrices associated with the exponential families, Linear algebra appl., 318, 1-3, 45-52, (2000) · Zbl 0964.15028 [46] Zhao, X.; Wang, T., Some identities related to reciprocal functions, Discrete math., 265, 1-3, 323-335, (2003) · Zbl 1017.05022
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