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...