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Good afternoon, everyone. Does anyone know what the Riemann $P$-symbols mean when they contain more than three columns (i.e., to what ordinary differential equations they correspond)?

Examples: $P\left\{\begin{array}{ccccc}0&1&a&b&\infty\\ 0&-1/2&-1/2&-1/2&2\\ 0&-1/2&1/2&1/2&2\\ \end{array} \zeta \right\}$; $P\left\{\begin{array}{ccccc}0&1&a&b&\infty\\ 0&-3/4&-1/4&-1/2&2\\ 0&-1/2&1/2&1/2&2\\ \end{array} \zeta \right\}$.

Appropriate references are encouraged.

Context of the question: the former of these symbols represent the known (textbook) solution for water flow in the soil; it is transformed as $$ \begin{array}{l} P\left\{\begin{array}{ccccc}0&1&a&b&\infty\\ 0&-1/2&-1/2&-1/2&2\\ 0&-1/2&1/2&1/2&2\\ \end{array} \zeta \right\}= (1-\zeta)^{-1/2}(\zeta-a)^{-1/2}(\zeta-b)^{-1/2}P\left\{\begin{array}{ccccc}0&1&a&b&\infty\\ 0&0&0&0&1/2\\ 0&0&1&1&1/2\\ \end{array} \zeta \right\} =\\ \qquad \qquad \qquad = (1-\zeta)^{-1/2}(\zeta-a)^{-1/2}(\zeta-b)^{-1/2}P\left\{\begin{array}{ccc}0&1&\infty\\ 0&0&1/2\\ 0&0&1/2\\ \end{array} \zeta \right\} \end{array} $$ without explanation of the operations fulfilled. The latter symbol, supposingly, represents another flow of a similar class that is of interest to me.

Some background I've found thus far: V.I. Smirnov relates the general three-column symbol $P\left\{\begin{array}{ccc}0&1&\infty\\ \alpha_1&\beta_1&\gamma_1;\\ \alpha_2&\beta_2&\gamma_2\\ \end{array} x \right\}$ to the equation $$ \begin{array}{l} w''+\left[\frac{1-(\alpha_1+\alpha_2)}z +\frac{1-(\beta_1+\beta_2)}{z-1} \right]w'+\\ \qquad +\left[\frac{\alpha_1\alpha_2}{z^2} +\frac{\beta_1\beta_2}{(z-1)^2} +(\gamma_1(\gamma_1-1) +(\alpha_1+\alpha_2+\beta_1+\beta_2)\gamma_1 +\alpha_1\alpha_2 +\beta_1\beta_2)\left(\frac 1z-\frac 1{z-1}\right) \right]w=0. \end{array} $$ Bateman, Erdelyi relate the four-column symbol of a specific type $P\left\{\begin{array}{cccc}0&1&a&\infty\\ 0&0&0&\alpha\\ 1-\gamma&1-\delta&1-\epsilon&\beta\\ \end{array} x \right\}$ to the equation $$ \frac{d^2w}{dx^2} +\left(\frac \gamma x+\frac \delta{x-1} +\frac \epsilon{x-a}\right)\frac{dw}{dx} +\frac{\alpha\beta x-q}{x(x-1)(x-a)}w=0. $$ This does not allow to "read" the ones of interest, though...

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Riemann symbol labels a class of differential equations of second order with regular singularities (which is the same as Fuchsian equations). The first row lists the singularities the second and third row list exponents at each singularity. The singularities and the exponents are not ordered so the class does not change when you permute columns, and when you permute the second and third entry in any column. The exponents must satisfy the Fuchs condition $$\sum (\lambda_i^\prime+\lambda_i^{\prime\prime})=n-2,$$ where $n$ is the number of singularities (=number of columns). Otherwise they are arbitrary.

This symbol was used to Riemann to represent an equation with three singularities (3 columns). In this case it defines the equation uniquely. In the case of 4 or more columns the symbol does not define the equation uniquely but defines a whole class of equations. To define an equation uniquely one has to specify additional $n-3$ parameters which are called accessory parameters. They are not reflected in the Riemann symbol. The general form of a Fuchsian equation with given Riemann symbol is written, for example on the first page 127 (in the bottom, formula (1)) of this paper:

Edward Van Vleck, On certain differential equations of second order allied to Hermite's equation, Amer. J. Math., 21 (1899) 2, 126-167. (available free on Internet).

Accessory parameters are denoted in this formula by $a_1,\ldots,a_{r-2}$. This paper gives a good example of using the Riemann symbol with 4 columns.

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    $\begingroup$ Excuse me, does the formula (1) in Edward Van Vleck you mentioned really read $$ \frac{d^2y}{dx^2} +\left[\sum \limits_{i=1}^r \frac{1-(\lambda_i'+\lambda_i'')}{x-e_i} \right]\frac{dy}{dx} +\left[\sum \limits_{i=1}^r \frac{\lambda_i'\lambda_i''}{(x-e_i)^2} +\frac{\lambda_\infty'\lambda_\infty'' -\sum \limits_{i=1}^r \lambda_i'\lambda_i'' x^{r-2}+a_1x^{r-3}+\ldots +a_{r-2}}{\prod \limits_{i=1}^r (x-e_i)} \right]y=0. $$ $\endgroup$ – Pavel Mostovykh Aug 17 at 15:04
  • $\begingroup$ or there is a misprint, and it should be $$ \frac{d^2y}{dx^2} +\left[\sum \limits_{i=1}^r \frac{1-(\lambda_i'+\lambda_i'')}{x-e_i} \right]\frac{dy}{dx} +\left[\sum \limits_{i=1}^r \frac{\lambda_i'\lambda_i''}{(x-e_i)^2} +\frac{\left(\lambda_\infty'\lambda_\infty'' -\sum \limits_{i=1}^r \lambda_i'\lambda_i''\right) x^{r-2}+a_1x^{r-3}+\ldots +a_{r-2}}{\prod \limits_{i=1}^r (x-e_i)} \right]y=0. $$ (parenthesis added for the $x^{r-2}$ term)? $\endgroup$ – Pavel Mostovykh Aug 17 at 15:05
  • $\begingroup$ Yes, there is a misprint, and you wrote it correctly. Sorry: I knew about this misprint, but forgot when I was answering your question. $\endgroup$ – Alexandre Eremenko Aug 17 at 17:07

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