# A particular case of Truesdell's unified theory of special functions

I'm reading through Clifford Truesdell's "An essay toward a unified theory of special functions", Princeton Univ. Press, 1948. All his exposition is based on the functional equation

$$\frac{\partial}{\partial z}\mathrm F(z,\alpha)=\mathrm F(z,\alpha+1)$$ He starts with

We are going to study functions $f (y, \alpha)$ satisfying a functional equation of the type

$$\frac{\partial}{\partial y} f (y, \alpha) = \mathrm A(y, \alpha) f (y, \alpha) + \mathrm B(y, a) f (y, \alpha+1 )$$

Then, we define

$$g\left( {y,\alpha } \right) = f\left( {y,\alpha } \right)\exp \left\{ { - \int\limits_{{y_0}}^y {\mathrm A\left( {v,\alpha } \right)dv} } \right\}$$

We verify that $g$ satisfies

$$\frac{\partial }{{\partial y}}g\left( {y,\alpha } \right) = g\left( {y,\alpha + 1} \right)B\left( {y,\alpha } \right)\exp \left\{ { - \int\limits_{{y_0}}^y {\left[ {A\left( {v,\alpha + 1} \right) - A\left( {v,\alpha } \right)} \right]dv} } \right\}$$

Thus we reduce the equation to

$$\frac{\partial }{{\partial y}}g\left( {y,\alpha } \right) = C\left( {y,\alpha } \right)g\left( {y,\alpha + 1} \right)\tag {1}$$

Now he states

In the case of nearly every special function that I know to satisfy an equation of type $(1)$, the coefficient $C(y, \alpha)$ is factorable, $C(y, \alpha)=Y(y)A( \alpha)$, so we asume

$$\frac{\partial }{{\partial y}}g\left( {y,\alpha } \right) = Y(y)A( \alpha)g\left( {y,\alpha + 1} \right)$$

Now he defines:

$$z:= \int_{y_1}^y Y(v) dv$$

and

$$F(z,\alpha ): = g\left( {y,\alpha } \right)\exp \left\{ {\mathop {\mathrm S}\limits_{{\alpha _0}}^\alpha \log {\text{A}}\left( v \right)\Delta v} \right\}$$

Now this is the operator that is troubling me

$$\mathop {\mathrm S}\limits_{{\alpha _0}}^\alpha h\left( v \right)\Delta v = \mathop {\lim }\limits_{k \to {0^ + }} \left\{ {\int\limits_{{a_0}}^\infty {h\left( v \right){e^{ - kc\left( v \right)}}dv} - \sum\limits_{m = 0}^\infty {h\left( {a + m} \right){e^{ - kc\left( {a + m} \right)}}} } \right\}$$

I can't find any reference to what $c(v)$ is. Is this known operator? What is $c$?

Anyways, I have a simple case I need to transform:

Let $$\mathrm F\left( {x,\alpha } \right) = \int\limits_0^x {{{\left( {\frac{t}{{t + 1}}} \right)}^\alpha }} \frac{{dt}}{t}$$

Then we have the functional equation

$$\frac{\alpha }{x} \mathrm F\left( {x,\alpha } \right) - \frac{\alpha }{x} \mathrm F\left( {x,\alpha + 1} \right) = \frac{\partial }{{\partial x}} \mathrm F\left( {x,\alpha } \right)$$

Following Truesdell's method, I define

$$\mathrm G\left( {x,\alpha } \right) = \frac{{\mathrm F\left( {x,\alpha } \right)}}{{{x^\alpha }}}$$

Then I have the functional equation

$$\frac{\partial }{{\partial x}} \mathrm G\left( {x,\alpha } \right) = - \alpha \mathrm G\left( {x,\alpha + 1} \right)$$

How can I transform it to the $\mathrm F$ equation using Truesdell's method?

The importance of the original $\mathrm F$ I define is that it can be used to show that

$$\log (1+x)=\sum_{n=1}^\infty \frac{1}{n}\left(\frac x {x+1} \right)^n\text{ ; for } x > -\frac 1 2$$

and maybe some other results can be derived. I still have a lot of exposition to read.

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The expression with the puzzling $\rm\:c(v)\:$ is Norlund's principal solution of the difference equation $\rm \mathop\Delta\limits_{\alpha}\ \mathop {\mathrm S}\limits_{{\alpha _0}}^\alpha h(v) dv = h(a).\:$ As Truesdell mentions in Appendix II, one can find an exposition of this in Chapter 8 of the classic The Calculus of Finite Differences by Milne-Thomson.
Bill, could you show how to evaluate $$\mathop {\mathrm S}\limits_{{\alpha _0}}^\alpha h\left( v \right)\Delta v = \mathop {\lim }\limits_{k \to {0^ + }} \left\{ {\int\limits_{{a_0}}^\infty {h\left( v \right){e^{ - kc\left( v \right)}}dv} - \sum\limits_{m = 0}^\infty {h\left( {a + m} \right){e^{ - kc\left( {a + m} \right)}}} } \right\}$$ for my special case? – Pedro Tamaroff Jun 9 '12 at 19:10
@Peter After decades, I don't recall the analytic technicalities of Norlund's method of summation. Besides Milne-Thompson, there are probably many expositions that can be found by googling "Norlund principal solution". However, when first learning Truesdell's method, I recommend first concentrating on the formal viewpoint, delaying the analytic complications till later. So, in your example, simply consider the solution of the difference equation as a formal sum $\rm\:\sum_{m = 0}^\infty h(a + m)\:.\:$ The other terms are introduced only to obtain convergence more widely. – Bill Dubuque Jun 9 '12 at 20:47