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Next: Gauss-Meixner Summation Up: Triangular Input Balanced Realizations Previous: Triangular Input Balanced Realizations

Meixner Functions and TIB Systems

The Meixner polynomials tex2html_wrap_inline1326 (Cf. The Bateman Manuscript Project: Higher Transcendental Functions, v. 2) can be defined by the generating function
equation616
convergent for tex2html_wrap_inline1328. In the following, we restrict ourselves to the case tex2html_wrap_inline1330, otherwise the Meixner polynomials do not correspond to linear time-invariant filters. Therefore we write tex2html_wrap_inline1332 instead of tex2html_wrap_inline1326. We define the sequence of weights tex2html_wrap_inline1336, then the Meixner polynomials satisfy the orthogonality relation
equation633
which suggests the definition of the Meixner functions
equation645
where
equation655
normalizes the functions to
equation664
Note that tex2html_wrap_inline1338.

We define the system
eqnarray682
where
equation684
and the initial conditions
equation695
This system has the solution
equation704
The system tex2html_wrap_inline1280 is TIB by the orthonormality of the Meixner polynomials. Another way to observe this by plugging tex2html_wrap_inline1280 back into the Stein equation.


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Fri Jun 27 03:10:38 EDT 1997

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