The Truesdell Method

  • Elna Browning McBride
Part of the Springer Tracts in Natural Philosophy book series (STPHI, volume 21)

Abstract

Truesdell’s study of the functional equation
$$ \frac{\partial }{{{\partial_{{\rm Z}}}}}\,F\left( {z,\alpha } \right) = F\left( {z,\,\alpha + 1} \right), $$
called the F-equation, yielded many valuable results, among which is his method for obtaining generating functions. By this method Truesdell [1] obtained both ascending and descending generating functions. For example, if the elements of a given set of functions \( \left\{ {{\phi_n}(x)} \right\} \) satisfy a differential-difference equation of the ascending type,
$$ \phi_n^{'}\,(x) = A\left( {x,\,n} \right)\,{\phi_n}\,(x) + B\left( {x,\,n} \right)\,{\phi_{{n + 1}}}\,(x), $$
with coefficients A(x, n) and B(x, n) suitably restricted, then a generating function can be found for the set \( \left\{ {{\phi_{{k + n}}}\,(x)} \right\}, \), where k is fixed. Furthermore, if the elements of the set satisfy a differential-difference equation of the descending type,
$$ \phi_n^{'}\,(x) = C\left( {x,\,n} \right)\,{\phi_n}\,(x) + D\left( {x,\,n} \right)\,{\phi_{{n - 1}}}\,(x), $$
with coefficients suitably restricted, a generating function can be found for the set \( \left\{ {{\phi_{{k - n}}}\,(x)} \right\}, \), when k is fixed.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • Elna Browning McBride
    • 1
  1. 1.Memphis State UniversityMemphisUSA

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