# 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.