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Generating functions for generalized binomial distributions

### Abstract

In a recent article generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal probability distributions was a key point to allow to give them a statistical interpretation in terms of probabilities. In this article we present an approach based on generating functions that solves the previous difficulties. Our main theorem makes explicit the conditions under which those formal probability distributions are always non-negative. Therefore, the constraints of non-negativeness are automatically fulfilled giving a complete characterization in terms of generating functions. A large number of analytical examples becomes available.

© 2012 American Institute of Physics

Received 22 May 2012
Accepted 14 September 2012
Published online 18 October 2012

Acknowledgments:
E.M.F.C. acknowledges the partial financial supports by CNPq, CAPES, and FAPERJ (Brazilian scientific agencies).

Article outline:

I. INTRODUCTION
II. THE GENERALIZED BINOMIAL DISTRIBUTION AND ITS GENERATING FUNCTIONS
A. The binomial-like distribution
B. The generating function point of view
1. The generating functions set Σ for the *x* _{ n }!’s
2. The generating functions set Σ_{+} for sequences of complete statistical type
C. The characterization of Σ_{+}
D. Internal deformations acting on Σ_{+}
1. The deformation
2. The deformation
3. Deformed-related transformations in
4. Generating new functions of from a known function of
5. The η deformation
III. EXAMPLES
A. Example 1
1. The case *n* = 2: An explicit simple example presenting correlations
B. Example 2
1. Finite sequence class
2. Infinite sequence class
C. Example 3: Nontrivial case with infinite radius of convergence
1. Relation with Hermite polynomials and calculation of *x* _{ n }
2. Asymptotic behavior of *x* _{ n }
3. Polynomials
IV. POSSIBLE APPLICATIONS
V. CONCLUSION

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2012-10-18

2016-02-07

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