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Database: ingentaconnect
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TY - ABST
AU - Blasiak, P.
AU - Penson, K.A.
AU - Solomon, A.I.
TI - Dobinski-type relations and the log-normal distribution
JO - Journal of Physics A: Mathematical and General
PY - 2003-01-01T00:00:00///
VL - 36
IS - 18
SP - L273
EP - L278
N2 -

We consider sequences of generalized Bell numbers *B*(*n*), *n* 1, 2, ..., which can be represented by Dobinski-type summation formulae, i.e. *B*(*n*) 1/*C*_{k 0}^{} [*P*(*k*)]^{n}/*D*(*k*), with *P*(*k*) a polynomial, *D*(*k*) a function of *k* and *C* const. They include the standard Bell numbers (*P*(*k*) *k*, *D*(*k*) *k*!, *C**e*), their generalizations *B*_{r,r}(*n*), *r* 2, 3, ..., appearing in the normal ordering of powers of boson monomials (*P*(*k*) (*k*+*r*)!/*k*!, *D*(*k*) *k*!, *C**e*), variants of 'ordered' Bell numbers *B*_{o}^{(p)}(*n*) (*P*(*k*) *k*, *D*(*k*) (*p*+1/*p*)^{k}, *C* 1 + *p*, *p* 1, 2 ...), etc. We demonstrate that for , , , *t* positive integers (, *t* 0), [*B*(*n*^{2} + *n* + )]^{t} is the *n*th moment of a positive function on (0, ) which is a weighted infinite sum of log-normal distributions.
UR - http://www.ingentaconnect.com/content/iop/jphysa/2003/00000036/00000018/art00101
ER -