# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A symbolic operator approach to several summation formulas for power series. II. (English) Zbl 1152.05304

Summary: [For part I, see He, T.X., Hsu, L.C., Shiue, P.J.-S., and Torney, D.C., J. Comput. Appl. Math. 177, No. 1, 17–33 (2005; Zbl 1064.65002).]

A kind of symbolic operator method is expanded here that can be used to construct many transformation formulas and summation formulas for various types of power series including some old ones and more new ones.

##### MSC:
 05A15 Exact enumeration problems, generating functions 65B10 Summation of series (numerical analysis) 33C45 Orthogonal polynomials and functions of hypergeometric type 39A70 Difference operators 41A80 Remainders in approximation formulas
##### References:
 [1] Comtet, L.: Advanced combinatorics, (1974) [2] G.P. Egorychev, Integral Representation and the Computation of Combinatorial Sums, Translation of Mathematical Monographs, vol. 59, American Mathematical Society, Providence, RI, 1984. [3] Gould, H. W.: Inverse series relations and other expansions involving Humbert polynomials, Duke math. J. 32, 697-711 (1965) · Zbl 0135.12001 · doi:10.1215/S0012-7094-65-03275-8 [4] Gould, H. W.: Combinatorial identities, (1972) [5] Gould, H. W.; Wetweerapong, J.: Evaluation of some classes of binomial identities and two new sets of polynomials, Indian J. Math. 41, No. 2, 159-190 (1999) · Zbl 1034.05002 [6] He, T. X.; Hsu, L. C.; Shiue, P. J. -S.: On the convergence of the summation formulas constructed by using a symbolic operator approach, Comput. math. Appl. 51, No. 3 – 4, 441-450 (2006) · Zbl 1228.65005 · doi:10.1016/j.camwa.2005.10.006 [7] He, T. X.; Hsu, L. C.; Shiue, P. J. -S.: Symbolization of generating functions, an application of Mullin – Rota’s theory of binomial enumeration, Comput. math. Appl. 54, No. 5, 664-678 (2007) · Zbl 1155.65300 · doi:10.1016/j.camwa.2006.12.034 [8] He, T. X.; Hsu, L. C.; Shiue, P. J. -S.; Torney, D. C.: A symbolic operator approach to several summation formulas for power series, J. comput. Appl. math. 177, 17-33 (2005) · Zbl 1064.65002 · doi:10.1016/j.cam.2004.08.002 [9] Howard, F. T.: Degenerate weighted Stirling numbers, Discrete math. 57, No. 1, 45-58 (1985) · Zbl 0606.10009 · doi:10.1016/0012-365X(85)90155-4 [10] Hsu, L. C.; Shiue, P. J. -S.: Cycle indicators and special functions, Ann. combin. 5, 179-196 (2001) · Zbl 0987.05007 · doi:10.1007/PL00001299 [11] Jolley, L. B. W.: Summation of series, (1961) · Zbl 0101.28602 [12] Jordan, Ch.: Calculus of finite differences, (1965) · Zbl 0154.33901 [13] Petkovšek, M.; Wilf, H. S.; Zeilberger, D.: A=B, (1996) [14] Roman, S.; Rota, G. -C.: The umbral calculus, Adv. math., 95-188 (1978) · Zbl 0375.05007 · doi:10.1016/0001-8708(78)90087-7 [15] Sofo, A.: Computational techniques for the summation of series, (2003) [16] Wang, X. -H.; Hsu, L. C.: A summation formula for power series using Eulerian fractions, Fibonacci quart. 41, No. 1, 23-30 (2003) · Zbl 1027.11013 [17] Wilf, H. S.: Generatingfunctionology, (1994) · Zbl 0831.05001