×

zbMATH — the first resource for mathematics

A determinantal approach to Appell polynomials. (English) Zbl 1200.33020
Families of Appell polynomials \((A_n(x))_{n\geq 0}\) satisfy \(\frac{\mathrm{d} A_n(x)}{\mathrm{d} x}=n A_{n-1}(x)\) for \(n\geq 1\). In the present paper a way is presented to express Appell polynomials as certain determinants. From this representation (and by using some linear algebra) the authors get several general properties of Appell polynomials such as recurrences, addition and multiplication theorems, forward differences and symmetry properties. An automated way is presented how to get the coefficients of these polynomials. In the final section, some classical examples of Appell polynomials are revisited.

MSC:
33C65 Appell, Horn and Lauricella functions
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
11B83 Special sequences and polynomials
65F40 Numerical computation of determinants
Software:
mctoolbox
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Appell, P., Sur une classe de polynomes, Annales scientifique de l’E.N.S., s. 2, 9, 119-144, (1880) · JFM 12.0342.02
[2] J. Bernoulli, Ars Conjectandi, Basel, 1713, p. 97.
[3] Jordan, C., Calculus of finite differences, (1965), Chelsea Pub. Co. New York · Zbl 0154.33901
[4] Euler, L., Methodus generalis summandi progressiones, Commentarii academiae scientiarum petropolitanae, 6, (1738)
[5] Boas, R.P., Polynimial expansions of analytic functions, (1964), Springer-Verlag Berlin, Gottingen, Heidelberg
[6] Bretti, G.; Cesarano, C.; Ricci, P.E., Laguerre-type exponentials and generalized Appell polynomials, Computers and mathematics with applications, 48, 833-839, (2004) · Zbl 1072.33010
[7] Bretti, G.; Natalini, P.; Ricci, P.E., Generalizations of the Bernoulli and Appell polynomials, Abstract and applied analysis, 7, 613-623, (2004) · Zbl 1072.33005
[8] Sheffer, I.M., A differential equation for Appell polynomials, American mathematical society. bulletin, 41, 914-923, (1935) · Zbl 0013.16602
[9] Shohat, J., The relation of the classical orthogonal polynomials to the polynomials of Appell, Americal journal of mathematics, 58, 453-464, (1936) · Zbl 0014.30802
[10] Roman, S., The umbral calculus, (1984), Academic Press New York · Zbl 0536.33001
[11] Douak, K., The relation of the d-orthogonal polynomials to the Appell polynomials, Journal of computation and applied mathematics, 70, 279-295, (1996) · Zbl 0863.33007
[12] He, M.X.; Ricci, P.E., Differential equation of Appell polynomials via the factorization method, Journal of computational and applied mathematics, 139, 231-237, (2002) · Zbl 0994.33008
[13] Dattoli, G.; Ricci, P.E.; Cesarano, C., Differential equations for Appell type polynomials, Fractional calculus & applied analysis, 5, 69-75, (2002) · Zbl 1040.33007
[14] Bretti, G.; He, M.X.; Ricci, P.E., On quadrature rules associated with Appell polynomials, International journal of applied mathematics, 11, 1-14, (2002) · Zbl 1041.65025
[15] Ismail, M.E.H., Remarks on differential equation of Appell polynomials via the factorization method, Journal of computational and applied mathematics, 154, 243-245, (2003)
[16] Bretti, G.; Ricci, P.E., Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese journal of mathematics, 8, 3, 415-428, (2004) · Zbl 1084.33007
[17] Costabile, F.; Dell’Accio, F.; Gualtieri, M.I., A new approach to Bernoulli polynomials, Rendiconti di matematica, S VII, 26, 1-12, (2006) · Zbl 1105.11002
[18] Highman, N.H., Accuracy and stability of numerical algorithms, (1996), SIAM Philadelphia
[19] Abramowitz, M.; Stegun, I., Handbook of mathematical functions, (1964), Dover New York · Zbl 0171.38503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.