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Coxeter group actions and limits of hypergeometric series

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Abstract

In this paper, we use combinatorial group theory and a limiting process to connect various types of hypergeometric series, and of relations among such series. We begin with a set S of 56 distinct translates of a certain function M, which takes the form of a Barnes integral, and is expressible as a sum of two very-well-poised \(_9F_8\) hypergeometric series of unit argument. We consider a known, transitive action of the Coxeter group \(W(E_7)\) on this set. We show that, by removing from \(W(E_7)\) a particular generator, we obtain a subgroup that is isomorphic to \(W(D_6)\), and that acts intransitively on S, partitioning it into three orbits, of sizes 32, 12, and 12, respectively. Taking certain limits of the M functions in the first orbit yields a set of 32 J functions, each of which is a sum of two Saalschützian \(_4F_3\) hypergeometric series of unit argument. The original action of \(W(D_6)\) on the M functions in this orbit is then seen to correspond to a known action of this group on this set of J functions. In a similar way, the image of each of the size-12 orbits, under a similar limiting process, is a set of 12 L functions that have been investigated in earlier works. In fact, these two image sets are the same. The limiting process is seen to preserve distance, except on pairs consisting of one M function from each size-12 orbit. Finally, each known three-term relation among the J and L functions is seen to be obtainable as a limit of a known three-term relation among the M functions.

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References

  1. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  2. Barnes, E.W.: A new development of the theory of hypergeometric functions. Proc. Lond. Math. Soc. 2(6), 141–177 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barnes, E.W.: A transformation of generalized hypergeometric series. Q. J. Math. 41, 136–140 (1910)

    MATH  Google Scholar 

  4. Beyer, W.A., Louck, J.D., Stein, P.R.: Group theoretical basis of some identities for the generalized hypergeometric series. J. Math. Phys. 28(3), 497–508 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bump, D.: Barnes’ second lemma and its application to Rankin–Selberg convolutions. Am. J. Math. 110(1), 179–185 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Drake, G. (ed.): Springer Handbook of Atomic. Molecular and Optical Physics. Springer, New York (2006)

    Google Scholar 

  7. Formichella, M., Green, R.M., Stade, E.: Coxeter group actions on \({}_4F_3(1)\) hypergeometric series. Ramanujan J. 24(1), 93–128 (2011)

    MathSciNet  MATH  Google Scholar 

  8. Gauss, C.F.: Disquisitiones generales circa seriem infinitam \(1+\frac{\alpha \beta }{1 \times \gamma }x +\frac{\alpha (\alpha +1)\beta (\beta +1)}{1 \times 2 \times \gamma (\gamma +1)}xx +\text{etc.}\), Werke 3. Königliche Gesellschaft der Wissenschaften, Göttingen, pp. 123–162 (1876)

  9. Green, R.M.: Combinatorics of Minuscule Representations. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  10. Green, R.M., Mishev, I., Stade, E.: Relations among complementary and supplementary pairings of Saalschützian \({}_4F_3(1)\) series. Ramanujan J. 39(3), 647–679 (2016)

    MathSciNet  MATH  Google Scholar 

  11. Grozin, A.: Lectures on QED and QCD: Practical Calculation and Renormalization of One- and Multi-Loop Feynman Diagrams. World Scientific, Singapore (2007)

    Book  MATH  Google Scholar 

  12. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  13. Lievens, S., Van der Jeugt, J.: Invariance groups of three term transformations for basic hypergeometric series. J. Comput. Appl. Math. 197(1), 1–14 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lievens, S., Van der Jeugt, J.: Symmetry groups of Bailey’s transformations for \({}_{10}\phi _9\)-series. J. Comput. Appl. Math. 206(1), 498–519 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Mishev, I.: Coxeter group actions on Saalschützian \({}_4F_3(1)\) series and very-well-poised \({}_7F_6(1)\) series. J. Math. Anal. Appl. 385(2), 1119–1133 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Raynal, J.: On the definition and properties of generalized 6-\(j\) symbols. J. Math. Phys. 20(12), 2398–2415 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Roy, M.: Coxeter group actions on complementary pairs of very well-poised \({}_9 F_8(1)\) hypergeometric series, Ph.D. Thesis, University of Colorado (2011)

  18. Stade, E.: Hypergeometric series and Euler factors at infinity for \(L\)-functions on \(GL(3,\mathbb{R}) \times GL(3,\mathbb{R})\). Am. J. Math. 115(2), 371–387 (1993)

    MathSciNet  MATH  Google Scholar 

  19. Stade, E.: Mellin transforms of Whittaker functions on \(GL(4,\mathbb{R})\) and \(GL(4,\mathbb{C})\). Manuscr. Math. 87(4), 511–526 (1995)

    MathSciNet  MATH  Google Scholar 

  20. Stade, E.: Mellin transforms of \(GL(n,\mathbb{R})\) Whittaker functions. Am. J. Math. 123(1), 121–161 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Stade, E.: Archimedean \(L\)-factors on \(GL(n) \times GL(n)\) and generalized Barnes integrals. Israel J. Math. 127, 201–220 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Stade, E., Taggart, J.: Hypergeometric series, a Barnes-type lemma, and Whittaker functions. J. Lond. Math. Soc. 61(1), 177–196 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thomae, J.: Ueber die Funktionen welche durch Reihen der Form dargestellt werden: \(1+\frac{pp^{\prime }p^{\prime \prime }}{1q^{\prime }q^{\prime \prime }}+ \cdots \). J. Reine Angew. Math. 87, 26–73 (1879)

    MathSciNet  MATH  Google Scholar 

  24. Titchmarsh, E.C.: The Theory of Functions. Oxford University Press, London (1952)

    Google Scholar 

  25. van de Bult, F.J., Rains, E.M., Stokman, J.V.: Properties of generalized univariate hypergeometric functions. Commun. Math. Phys. 275(1), 37–95 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Whipple, F.J.W.: A group of generalized hypergeometric series: relations between 120 allied series of the type \(F(a, b, c;e, f)\). Proc. Lond. Math. Soc. 23(2), 247–263 (1925)

    MathSciNet  Google Scholar 

  27. Whipple, F.J.W.: Well-poised series and other generalized hypergeometric series. Proc. Lond. Math. Soc. 25(2), 525–544 (1926)

    Article  MATH  Google Scholar 

  28. Whipple, F.J.W.: Relations between well-poised hypergeometric series of the type \({}_7F_6\). Proc. Lond. Math. Soc. 40(2), 336–344 (1936)

    MATH  Google Scholar 

  29. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1963)

    MATH  Google Scholar 

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  1. University of Colorado, Boulder, CO, USA

    R. M. Green, Ilia D. Mishev & Eric Stade

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  1. R. M. Green
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  2. Ilia D. Mishev
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  3. Eric Stade
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Appendix: Explicit form of elements of the orbits \(\pmb {\mathscr {O}}_1\), \(\pmb {\mathscr {O}}_2\), and \(\pmb {\mathscr {O}}_3\), with corresponding \(J/L\) functions

Appendix: Explicit form of elements of the orbits \(\pmb {\mathscr {O}}_1\), \(\pmb {\mathscr {O}}_2\), and \(\pmb {\mathscr {O}}_3\), with corresponding \(J/L\) functions

1.1 Orbit \(\pmb {\mathscr {O}}_1\) (blue L)

$$\begin{aligned} M_{v(0,7)}(\vec {w})&=M \biggl [{\begin{array}{ll}a;b;c,d,\\ e,f,g,h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_6(\vec {x})&=L\biggl [{\begin{array}{ll}e,f,g,1+a-c-d;\\ e+f+g-a;1+a-c,1+a-d\end{array}}\biggr ]; \end{aligned}$$
(A.1)
$$\begin{aligned} M_{v(0,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+b-a;c,c+d-a,\\ c+e-a,c+f-a,c+g-a,c+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_5(\vec {x})&=L\biggl [{\begin{array}{ll} c+e-a,c+f-a,c+g-a,1-d; \\ c+e+f+g-2a;1+c-a,1+c-d\end{array}}\biggr ];\end{aligned}$$
(A.2)
$$\begin{aligned} M_{v(0,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}2d-a;d+b-a;d,d+c-a,\\ d+e-a,d+f-a,d+g-a,d+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_4(\vec {x})&=L\biggl [{\begin{array}{ll}d+e-a,d+f-a,d+g-a,1-c;\\ d+e+f+g-2a;1+d-a,1+d-c\end{array}}\biggr ]\end{aligned}$$
(A.3)
$$\begin{aligned} M_{v(0,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}2e-a;e+b-a;e,e+c-a,\\ e+d-a,e+f-a,e+g-a,e+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_3(\vec {x})&=L\biggl [{\begin{array}{ll}e+d-a,e+f-a,e+g-a,1-c;\\ e+d+f+g-2a;1+e-a,1+e-c\end{array}}\biggr ] \end{aligned}$$
(A.4)
$$\begin{aligned} M_{v(0,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}2f-a;f+b-a;f,f+c-a,\\ f+d-a,f+e-a,f+g-a,f+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_2(\vec {x})&=L\biggl [{\begin{array}{ll}f+d-a,f+e-a,f+g-a,1-c;\\ f+d+e+g-2a;1+f-a,1+f-c\end{array}}\biggr ] \end{aligned}$$
(A.5)
$$\begin{aligned} M_{v(0,2)}(\vec {w})&=M\biggl [{\begin{array}{ll}2g-a;g+b-a;g,g+c-a,\\ g+d-a,g+e-a,g+f-a,g+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_1(\vec {x})&=L\biggl [{\begin{array}{ll}g+d-a,g+e-a,g+f-a,1-c;\\ g+d+e+f-2a;1+g-a,1+g-c\end{array}}\biggr ] \end{aligned}$$
(A.6)
$$\begin{aligned} M_{-v(1,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-h;1-c,1-d,\\ 1-e,1-f,1-g,1-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{6}}(\vec {x})&=L\biggl [{\begin{array}{ll}1-e,1-f,1-g,c+d-a;\\ 2+a-e-f-g;1+c-a,1+d-a\end{array}}\biggr ] \end{aligned}$$
(A.7)
$$\begin{aligned} M_{-v(1,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-h;1-c,1+a-c-d,\\ 1+a-c-e,1+a-c-f,1+a-c-g,1+a-c-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{5}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-c-e,1+a-c-f,1+a-c-g,d;\\ 2+2a-c-e-f-g;1+a-c,1+d-c\end{array}}\biggr ] ; \end{aligned}$$
(A.8)
$$\begin{aligned} M_{-v(1,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2d;1+a-d-h;1-d,1+a-d-c,\\ 1+a-d-e,1+a-d-f,1+a-d-g,1+a-d-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{4}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-d-e,1+a-d-f,1+a-d-g,c;\\ 2+2a-d-e-f-g;1+a-d,1+c-d\end{array}}\biggr ] \end{aligned}$$
(A.9)
$$\begin{aligned} M_{-v(1,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2e;1+a-e-h;1-e,1+a-e-c,\\ 1+a-e-d,1+a-e-f,1+a-e-g,1+a-e-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{3}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-e-d,1+a-e-f,1+a-e-g,c;\\ 2+2a-e-d-f-g;1+a-e,1+c-e\end{array}}\biggr ] \end{aligned}$$
(A.10)
$$\begin{aligned} M_{-v(1,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2f;1+a-f-h;1-f,1+a-f-c,\\ 1+a-f-d,1+a-f-e,1+a-f-g,1+a-f-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{2}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-f-d,1+a-f-e,1+a-f-g,c;\\ 2+2a-f-d-e-g;1+a-f,1+c-f\end{array}}\biggr ] \end{aligned}$$
(A.11)
$$\begin{aligned} M_{-v(1,2)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2g;1+a-g-h;1-g,1+a-g-c,\\ 1+a-g-d,1+a-g-e,1+a-g-f,1+a-g-b\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{1}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-g-d,1+a-g-e,1+a-g-f,c;\\ 2+2a-g-d-e-f;1+a-g,1+c-g\end{array}}\biggr ]. \end{aligned}$$
(A.12)

1.2 Orbit \(\pmb {\mathscr {O}}_2\) (red L)

$$\begin{aligned} M_{v(1,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;h;c,d,\\ e,f,g,b\end{array}}\biggr ]\nonumber \\ \leftrightarrow L_6(\vec {x})&= L\biggl [{\begin{array}{ll}e,f,g,1+a-c-d;\\ e+f+g-a;1+a-c,1+a-d\end{array}}\biggr ] \end{aligned}$$
(A.13)
$$\begin{aligned} M_{v(1,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+h-a;c,c+d-a,\\ c+e-a,c+f-a,c+g-a,c+b-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_5(\vec {x})&=L\biggl [{\begin{array}{ll}c+e-a,c+f-a,c+g-a,1-d;\\ c+e+f+g-2a;1+c-a,1+c-d\end{array}}\biggr ] \end{aligned}$$
(A.14)
$$\begin{aligned} M_{v(1,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}2d-a;d+h-a;d,d+c-a,\\ d+e-a,d+f-a,d+g-a,d+b-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_4(\vec {x})&=L\biggl [{\begin{array}{ll}d+e-a,d+f-a,d+g-a,1-c;\\ d+e+f+g-2a;1+d-a,1+d-c\end{array}}\biggr ] \end{aligned}$$
(A.15)
$$\begin{aligned} M_{v(1,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}2e-a;e+h-a;e,e+c-a,\\ e+d-a,e+f-a,e+g-a,e+b-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_3(\vec {x})&=L\biggl [{\begin{array}{ll}e+d-a,e+f-a,e+g-a,1-c;\\ e+d+f+g-2a;1+e-a,1+e-c\end{array}}\biggr ] ; \end{aligned}$$
(A.16)
$$\begin{aligned} M_{v(1,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}2f-a;f+h-a;f,f+c-a,\\ f+d-a,f+e-a,f+g-a,f+b-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_2(\vec {x})&=L\biggl [{\begin{array}{ll}f+d-a,f+e-a,f+g-a,1-c;\\ f+d+e+g-2a;1+f-a,1+f-c\end{array}}\biggr ] \end{aligned}$$
(A.17)
$$\begin{aligned} M_{v(1,2)}(\vec {w})&=M\biggl [{\begin{array}{ll}2g-a;g+h-a;g,g+c-a,\\ g+d-a,g+e-a,g+f-a,g+b-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_1(\vec {x})&=L\biggl [{\begin{array}{ll}g+d-a,g+e-a,g+f-a,1-c;\\ g+d+e+f-2a;1+g-a,1+g-c\end{array}}\biggr ] \end{aligned}$$
(A.18)
$$\begin{aligned} M_{-v(0,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-b;1-c,1-d,\\ 1-e,1-f,1-g,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{6}}(\vec {x})&=L\biggl [{\begin{array}{ll}1-e,1-f,1-g,c+d-a;\\ 2+a-e-f-g;1+c-a,1+d-a\end{array}}\biggr ] \end{aligned}$$
(A.19)
$$\begin{aligned} M_{-v(0,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-b;1-c,1+a-c-d,\\ 1+a-c-e,1+a-c-f,1+a-c-g,1+a-c-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{5}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-c-e,1+a-c-f,1+a-c-g,d;\\ 2+2a-c-e-f-g;1+a-c,1+d-c\end{array}}\biggr ] \end{aligned}$$
(A.20)
$$\begin{aligned} M_{-v(0,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2d;1+a-d-b;1-d,1+a-d-c,\\ 1+a-d-e,1+a-d-f,1+a-d-g,1+a-d-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{4}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-d-e,1+a-d-f,1+a-d-g,c;\\ 2+2a-d-e-f-g;1+a-d,1+c-d\end{array}}\biggr ] \end{aligned}$$
(A.21)
$$\begin{aligned} M_{-v(0,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2e;1+a-e-b;1-e,1+a-e-c,\\ 1+a-e-d,1+a-e-f,1+a-e-g,1+a-e-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{3}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-e-d,1+a-e-f,1+a-e-g,c;\\ 2+2a-e-d-f-g;1+a-e,1+c-e\end{array}}\biggr ] \end{aligned}$$
(A.22)
$$\begin{aligned} M_{-v(0,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2f;1+a-f-b;1-f,1+a-f-c,\\ 1+a-f-d,1+a-f-e,1+a-f-g,1+a-f-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{2}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-f-d,1+a-f-e,1+a-f-g,c;\\ 2+2a-f-d-e-g;1+a-f,1+c-f\end{array}}\biggr ] \end{aligned}$$
(A.23)
$$\begin{aligned} M_{-v(0,2)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2g;1+a-g-b;1-g,1+a-g-c,\\ 1+a-g-d,1+a-g-e,1+a-g-f,1+a-g-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow L_{\overline{1}}(\vec {x})&=L\biggl [{\begin{array}{ll}1+a-g-d,1+a-g-e,1+a-g-f,c;\\ 2+2a-g-d-e-f;1+a-g,1+c-g\end{array}}\biggr ]. \end{aligned}$$
(A.24)

1.3 Orbit \(\pmb {\mathscr {O}}_3\) (J)

$$\begin{aligned} M_{v(0,1)}(\vec {w})&=M\biggl [{\begin{array}{ll} 2+a-c-d-e-f;2+2a-c-d-e-f-g;1-c, 1-d, \\ 1-e,1-f,b+g-a,h+g-a\end{array}}\biggr ]\nonumber \\ \leftrightarrow J_{p_0}(\vec {x})&=J\biggl [{\begin{array}{ll}2+2a-c-d-e-f-g;1-c,1-d,1+a-c-d;\\ 2+a-c-d-e,2+a-c-d-f,2+a-c-d-g\end{array}}\biggr ]; \end{aligned}$$
(A.25)
$$\begin{aligned} M_{-v(3,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2e;1+a-e-f;1-e,1+a-e-b,\\ 1+a-e-c,1+a-e-d,1+a-e-g,1+a-e-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_1}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-e-f;1+a-e-c,1+a-e-d,g;\\ 2+2a-c-d-e-f,1+a-e,1+g-e\end{array}}\biggr ] \end{aligned}$$
(A.26)
$$\begin{aligned} M_{-v(2,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2e;1+a-e-g;1-e,1+a-e-b,\\ 1+a-e-c,1+a-e-d,1+a-e-f,1+a-e-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_2}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-e-g;1+a-e-c,1+a-e-d,f;\\ 2+2a-c-d-e-g,1+a-e,1+f-e\end{array}}\biggr ] \end{aligned}$$
(A.27)
$$\begin{aligned} M_{-v(2,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2f;1+a-f-g;1-f,1+a-f-b,\\ 1+a-f-c,1+a-f-d,1+a-f-e,1+a-f-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_3}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-f-g;1+a-f-c,1+a-f-d,e;\\ 2+2a-c-d-f-g,1+a-f,1+e-f\end{array}}\biggr ] \end{aligned}$$
(A.28)
$$\begin{aligned} M_{-v(6,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-c;1-b,1-d,\\ 1-e,1-f,1-g,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_4}(\vec {x})&=J\biggl [{\begin{array}{ll}1-c;1-d,1-e,f+g-a;\\ 2+a-c-d-e,1+f-a,1+g-a\end{array}}\biggr ] \end{aligned}$$
(A.29)
$$\begin{aligned} M_{v(2,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}2d-a;d+g-a;d,d+b-a,\\ d+c-a,d+e-a,d+f-a,d+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_5}(\vec {x})&=J\biggl [{\begin{array}{ll}d+g-a;d+c-a,d+e-a,1-f;\\ c+d+e+g-2a,1+d-a,1+d-f\end{array}}\biggr ] \end{aligned}$$
(A.30)
$$\begin{aligned} M_{v(3,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}2d-a;d+f-a;d,d+b-a,\\ d+c-a,d+e-a,d+g-a,d+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_6}(\vec {x})&=J\biggl [{\begin{array}{ll}d+f-a;d+c-a,d+e-a,1-g;\\ c+d+e+f-2a,1+d-a,1+d-g\end{array}}\biggr ] \end{aligned}$$
(A.31)
$$\begin{aligned} M_{v(4,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}2d-a;d+e-a;d,d+b-a,\\ d+c-a,d+f-a,d+g-a,d+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_7}(\vec {x})&=J\biggl [{\begin{array}{ll}d+e-a;d+c-a,d+f-a,1-g;\\ c+d+e+f-2a,1+d-a,1+d-g\end{array}}\biggr ] ; \end{aligned}$$
(A.32)
$$\begin{aligned} M_{-v(5,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-d;1-b,1-c,\\ 1-e,1-f,1-g,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_8}(\vec {x})&=J\biggl [{\begin{array}{ll}1-d;1-c,1-e,f+g-a;\\ 2+a-c-d-e,1+f-a,1+g-a\end{array}}\biggr ] \end{aligned}$$
(A.33)
$$\begin{aligned} M_{v(2,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+g-a;c,c+b-a,\\ c+d-a,c+e-a,c+f-a,c+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_9}(\vec {x})&=J\biggl [{\begin{array}{ll}c+g-a;c+d-a,c+e-a,1-f;\\ c+d+e+g-2a,1+c-a,1+c-f\end{array}}\biggr ] \end{aligned}$$
(A.34)
$$\begin{aligned} M_{v(3,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+f-a;c,c+b-a,\\ c+d-a,c+e-a,c+g-a,c+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_{10}}(\vec {x})&=J\biggl [{\begin{array}{ll}c+f-a;c+d-a,c+e-a,1-g;\\ c+d+e+f-2a,1+c-a,1+c-g\end{array}}\biggr ] \end{aligned}$$
(A.35)
$$\begin{aligned} M_{v(4,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+e-a;c,c+b-a,\\ c+d-a,c+f-a,c+g-a,c+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_{11}}(\vec {x})&=J\biggl [{\begin{array}{ll}c+e-a;c+d-a,c+f-a,1-g;\\ c+d+e+f-2a,1+c-a,1+c-g\end{array}}\biggr ] \end{aligned}$$
(A.36)
$$\begin{aligned} M_{-v(5,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-d;1-c,1+a-c-b,\\ 1+a-c-e,1+a-c-g,1+a-c-f,1+a-c-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_{12}}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-c-d;1+a-c-e,1+a-c-g,f;\\ 2+2a-c-d-e-g,1+a-c,1+f-c\end{array}}\biggr ] \end{aligned}$$
(A.37)
$$\begin{aligned} M_{v(2,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;g;b,c,\\ d,e,f,h\end{array}}\biggr ]\nonumber \\ \leftrightarrow J_{p_{13}}(\vec {x})&=J\biggl [{\begin{array}{ll}g;c,d,1+a-e-f;\\ c+d+g-a,1+a-e,1+a-f\end{array}}\biggr ] \end{aligned}$$
(A.38)
$$\begin{aligned} M_{v(3,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;f;b,c,\\ d,e,g,h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_{14}}(\vec {x})&=J\biggl [{\begin{array}{ll}f;c,d,1+a-e-g;\\ c+d+f-a,1+a-e,1+a-g\end{array}}\biggr ] \end{aligned}$$
(A.39)
$$\begin{aligned} M_{v(4,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;e;b,c,\\ d,f,g,h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{p_{15}}(\vec {x})&=J\biggl [{\begin{array}{ll}e;c,d,1+a-f-g;\\ c+d+e-a,1+a-f,1+a-g\end{array}}\biggr ] \end{aligned}$$
(A.40)
$$\begin{aligned} M_{-v(0,1)}(\vec {w})&=M\biggl [{\begin{array}{ll}c+d+e+f-a-1;c+d+e+f+g-2a-1;c,d, \\ e,f,1+a-b-g,1+a-h-g\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_0}(\vec {x})&=J\biggl [{\begin{array}{ll}c+d+e+f+g-2a-1;c,d,c+d-a;\\ c+d+e-a,c+d+f-a,c+d+g-a\end{array}}\biggr ] ; \end{aligned}$$
(A.41)
$$\begin{aligned} M_{v(3,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}2e-a;e+f-a;e,e+b-a,\\ e+c-a,e+d-a,e+g-a,e+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_1}(\vec {x})&=J\biggl [{\begin{array}{ll}e+f-a;e+c-a,e+d-a,1-g;\\ c+d+e+f-2a,1+e-a,1+e-g\end{array}}\biggr ] \end{aligned}$$
(A.42)
$$\begin{aligned} M_{v(2,4)}(\vec {w})&=M\biggl [{\begin{array}{ll}2e-a;e+g-a;e,e+b-a,\\ e+c-a,e+d-a,e+f-a,e+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_2}(\vec {x})&=J\biggl [{\begin{array}{ll}e+g-a;e+c-a,e+d-a,1-f;\\ c+d+e+g-2a,1+e-a,1+e-f\end{array}}\biggr ] \end{aligned}$$
(A.43)
$$\begin{aligned} M_{v(2,3)}(\vec {w})&=M\biggl [{\begin{array}{ll}2f-a;f+g-a;f,f+b-a,\\ f+c-a,f+d-a,f+e-a,f+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_3}(\vec {x})&=J\biggl [{\begin{array}{ll}f+g-a;f+c-a,f+d-a,1-e;\\ c+d+f+g-2a,1+f-a,1+f-e\end{array}}\biggr ] \end{aligned}$$
(A.44)
$$\begin{aligned} M_{v(6,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;c;b,d,\\ e,f,g,h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_4}(\vec {x})&=J\biggl [{\begin{array}{ll}c;d,e,1+a-f-g;\\ c+d+e-a,1+a-f,1+a-g\end{array}}\biggr ] ; \end{aligned}$$
(A.45)
$$\begin{aligned} M_{-v(2,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2d;1+a-d-g;1-d,1+a-d-b,\\ 1+a-d-c,1+a-d-e,1+a-d-f,1+a-d-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_5}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-d-g;1+a-d-c,1+a-d-e,f;\\ 1+2a-c-d-e-g,1+a-d,1+f-d\end{array}}\biggr ] \end{aligned}$$
(A.46)
$$\begin{aligned} M_{-v(3,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2d;1+a-d-f;1-d,1+a-d-b,\\ 1+a-d-c,1+a-d-e,1+a-d-g,1+a-d-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_6}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-d-f;1+a-d-c,1+a-d-e,g;\\ 1+2a-c-d-e-f,1+a-d,1+g-d\end{array}}\biggr ] \end{aligned}$$
(A.47)
$$\begin{aligned} M_{-v(4,5)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2d;1+a-d-e;1-d,1+a-d-b,\\ 1+a-d-c,1+a-d-f,1+a-d-g,1+a-d-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_7}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-d-e;1+a-d-c,1+a-d-f,g;\\ 1+2a-c-d-e-f,1+a-d,1+g-d\end{array}}\biggr ] \end{aligned}$$
(A.48)
$$\begin{aligned} M_{v(5,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}a;d;b,c,\\ e,f,g,h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_8}(\vec {x})&=J\biggl [{\begin{array}{ll}d;c,e,1+a-f-g;\\ c+d+e-a,1+a-f,1+a-g\end{array}}\biggr ] \end{aligned}$$
(A.49)
$$\begin{aligned} M_{-v(2,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-g;1-c,1+a-c-b,\\ 1+a-c-d,1+a-c-e,1+a-c-f,1+a-c-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_9}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-c-g;1+a-c-d,1+a-c-e,f;\\ 1+2a-c-d-e-f,1+a-c,1+f-c\end{array}}\biggr ] ; \end{aligned}$$
(A.50)
$$\begin{aligned} M_{-v(3,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-f;1-c,1+a-c-b,\\ 1+a-c-d,1+a-c-e,1+a-c-g,1+a-c-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{10}}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-c-f;1+a-c-d,1+a-c-e,g;\\ 1+2a-c-d-e-g,1+a-c,1+g-c\end{array}}\biggr ] \end{aligned}$$
(A.51)
$$\begin{aligned} M_{-v(4,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}1+a-2c;1+a-c-e;1-c,1+a-c-b,\\ 1+a-c-d,1+a-c-f,1+a-c-g,1+a-c-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{11}}(\vec {x})&=J\biggl [{\begin{array}{ll}1+a-c-e;1+a-c-d,1+a-c-f,g;\\ 1+2a-c-d-f-g,1+a-c,1+g-c\end{array}}\biggr ] ; \end{aligned}$$
(A.52)
$$\begin{aligned} M_{v(5,6)}(\vec {w})&=M\biggl [{\begin{array}{ll}2c-a;c+d-a;c,c+b-a,\\ c+e-a,c+g-a,c+f-a,c+h-a\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{12}}(\vec {x})&=J\biggl [{\begin{array}{ll}c+d-a;c+e-a,c+g-a,1-f;\\ c+d+e+g-2a,1+c-a,1+c-f\end{array}}\biggr ] \end{aligned}$$
(A.53)
$$\begin{aligned} M_{-v(2,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-g;1-b,1-c,\\ 1-d,1-e,1-f,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{13}}(\vec {x})&=J\biggl [{\begin{array}{ll}1-g;1-c,1-d,e+f-a;\\ 2+a-c-d-g,1+e-a,1+f-a\end{array}}\biggr ] \end{aligned}$$
(A.54)
$$\begin{aligned} M_{-v(3,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-f;1-b,1-c,\\ 1-d,1-e,1-g,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{14}}(\vec {x})&=J\biggl [{\begin{array}{ll}1-f;1-c,1-d,e+g-a;\\ 2+a-c-d-f,1+e-a,1+g-a\end{array}}\biggr ] \end{aligned}$$
(A.55)
$$\begin{aligned} M_{-v(4,7)}(\vec {w})&=M\biggl [{\begin{array}{ll}1-a;1-e;1-b,1-c,\\ 1-d,1-f,1-g,1-h\end{array}}\biggr ] \nonumber \\ \leftrightarrow J_{n_{15}}(\vec {x})&=J\biggl [{\begin{array}{ll}1-e;1-c,1-d,f+g-a;\\ 2+a-c-d-e,1+f-a,1+g-a\end{array}}\biggr ]. \end{aligned}$$
(A.56)

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Green, R.M., Mishev, I.D. & Stade, E. Coxeter group actions and limits of hypergeometric series. Ramanujan J 53, 607–651 (2020). https://doi.org/10.1007/s11139-020-00249-y

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