Multivariate Bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions. (English) Zbl 1162.05301
Summary: If a power series is raised to a power, the coefficients of the new power series are called the partial Bell polynomials. They are useful for obtaining the inverse power series, iterations (possibly fractional) of the original power series, logs and exponents of power series, the derivatives of a composite function, and for solving various problems in partition theory. Multivariate versions of these diverse and useful results are given in terms of multivariate Bell polynomials, introduced here for the first time.
Keywords:
Bell polynomials; differentiation of composite functions; fractionary iterates; inverse series; potential polynomials; powers of power seriesReferences:
| [1] | Aldrovandi, R., Special Matrices of Mathematical Physics (2001), World Scientific: World Scientific Singapore · Zbl 0982.15040 |
| [2] | Andrews, G. E., The theory of partitions, (Encyclopedia of Mathematics and Its Applications, vol. 2 (1976), Addison-Wesley) · Zbl 0155.09302 |
| [3] | Bell, E. T., Exponential polynomials, Annals of Mathematics, 35, 258-277 (1934) · Zbl 0009.21202 |
| [4] | Berndt, B. C., Ramanujan’s Notebooks, Part I (1985), Springer-Verlag: Springer-Verlag New York, With a foreword by S. Chandrasekhar · Zbl 0555.10001 |
| [5] | Blakley, G. R., Formal solution of nonlinear simultaneous equations: reversion of series in several variables, Duke Mathematical Journal, 31, 347-357 (1964) · Zbl 0122.01502 |
| [6] | Blasiak, P.; Gawron, A.; Horzela, A.; Penson, K. A.; Solomon, A. I., Exponential operators, dobinski relations and summability, Journal of Physics: Conference Series, 36, 22-27 (2006) · Zbl 1118.81054 |
| [7] | Blasiak, P.; Penson, K. A.; Solomon, A. I.; Horzela, A.; Duchamp, G. H.E., Some useful combinatorial formulas for bosonic operators, Journal of Mathemamtical Physics, 46, 0022-2488 (2005) · Zbl 1110.81112 |
| [8] | Comtet, L., Advanced Combinatorics (1974), Reidel: Reidel Dordrecht · Zbl 0283.05001 |
| [9] | Davies, R. B., Testing the hypothesis that a point process is Poisson, Advances in Applied Probability, 9, 724-746 (1977) · Zbl 0387.60055 |
| [10] | Pitman, J., Combinatorial stochastic processes, (Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard. Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7-24, 2002. With a foreword by Jean Picard, Lecture Notes in Mathematics, vol. 1875 (2006), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1103.60004 |
| [11] | Roman, S., The Umbral Calculus (1984), Academic Press, Inc.: Academic Press, Inc. New York · Zbl 0536.33001 |
| [12] | Sack, R. A., Generalisations of Lagrange’s expansion for functions of several implicity defined variables, Journal of SIAM, 13, 913-926 (1965) · Zbl 0143.27403 |
| [13] | Srivastava, H. M.; Manocha, H. L., A Treatise on Generating Functions (1984), John Wiley and Sons: John Wiley and Sons New York · Zbl 0535.33001 |
| [14] | Withers, C. S., Accurate confidence intervals for distributions with one parameter, Annals of the Institute of Statistical Mathematics A, 35, 49-61 (1983) · Zbl 0526.62030 |
| [15] | Withers, C. S., A chain rule for differentiation with applications to the multivariate Hermite polynomials, Bulletin of the Australian Mathematical Society, 30, 247-250 (1984) · Zbl 0549.33012 |
| [16] | C.S. Withers, Two multivariate multinomial theorems with applications to Hermite polynomials. Technical Report, Applied Mathematics Group, Industrial Research Ltd., Lower Hutt, New Zealand, 2008.; C.S. Withers, Two multivariate multinomial theorems with applications to Hermite polynomials. Technical Report, Applied Mathematics Group, Industrial Research Ltd., Lower Hutt, New Zealand, 2008. · Zbl 0648.62053 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
