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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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1 Answer

up vote 9 down vote accepted

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.

As I have mentioned previously here, Willard Miller showed that Truesdell's method is essentially Lie-theoretic. See his freely available book Lie theory and Special Functions, 1968. There he also shows that, similarly, the Schroedinger-Infeld-Hull ladder / factorization method (a powerful tool widely exploited by physicists to compute eigenvalues, recurrence relations, etc. for solutions of second order ODEs) is essentially equivalent to the representation theory of four local Lie groups. Nowadays it is a special case of Lie-theoretic symmetry methods used for separation of variables in partial differential equations (a major theme in the group-theoretic approach towards a unified theory of special functions).

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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
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@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
    
Hm, OK. Thank you very much! –  Pedro Tamaroff Jun 9 '12 at 20:50
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