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Problems and prospects for basic hypergeometric functions. (English) Zbl 0342.33001

Theory Appl. Spec. Funct., Proc. Adv. Semin., Madison 1975, 191-224 (1975).
In a recent survey paper [SIAM Rev. 16, 441–484 (1974; Zbl 0299.33004)], the author has discussed applications of the basic hypergeometric series \[ _m\phi_n \begin{bmatrix} a_1\ldots, a_n; q, z \\ b_1,\ldots, b_n\qquad\end{bmatrix} = \sum_{j=0}^\infty \frac{(a_1)_j \cdots (a_n)_j z^j}{(b_1)_j \cdots (b_n)_j (q)_j}, \] where \((a)_n = (a;q)_n = (1- a)(1 - aq) \cdots (1- aq^{n-1})\), \((a)_0 = 1\), to partitions, number theory, finite vector spaces, combinatorial identities and physics. The author notes that a number of recent results suggest that “the theory of basic hypergeometric functions can be greatly extended and that the resulting theory will have a considerable impact on the subjects that originally suggested the extension”.
In the present paper the author first describes the recent work in the theory of partitions that suggests the need for an extended theory of the basic hypergeometric series.
Problem 1. What finite linear homogeneous ordinary \(q\)-difference with coefficients that are polynomials in \(x\) and \(q\) have multiple basic hypergeometric series as solutions?
The author shows how one known case of Problem 1 leads to a proof of the following theorem of I. J. Schur [Sitzungsber. Akad. Wissensch. Berlin, Phys.-Math. Kl. 1926, 488–495 (1926; JFM 52.0166.03)]: The partitions of \(n\) into parts \(\equiv 1 \text{ or }5 \pmod 6\) are equinumerous with the partitions of \(n\) in which all parts differ by at least 3 and all parts that are multiples of 3 differ by at least 6.
Another example of a known case of Problem 1 is furnished by a result of A. Selberg [Avh. Norske Vid. Akad. Oslo 1936, No. 8, 1–23 (1936; Zbl 0016.24701, JFM 62.1068.04)]. In this connection the following identity is proved: \[ \sum_{n_1,\ldots, n_{k-1}\ge 0} \frac{q^{N_1^2+\ldots+N_{k-1}^2}}{(q)_{n_1}\cdots (q)_{n_{k-1}}} = \prod_{\substack{n=1 \\ n\not\equiv 0, \pm k(2k+1)}}^\infty (1-q^n)^{-1}, \tag{*} \] where \(N_j = n_j +n_{j+1}+\ldots+n_{k-1}\).
Problem 2. What are possible multiple series generalizations of the \(q\)-analog of Whipple’s theorem? G. N. Watson [J. Lond. Math. Soc. 4, 4–9 (1929; JFM 55.0219.09)] proved the following \(q\)-analog of Whipple’s theorem: \[ _6\phi_7 \begin{bmatrix} a, q\sqrt a, -q \sqrt a, b, c,d, e,q^{-N}; q, a^2q^{N+2}, bcde \\ \sqrt a, -\sqrt a, aq/b, aq/c, aq/d, aq/e, aq^{N+1}\qquad \end{bmatrix} = \tag{**} \] \[ = \frac{(aq)_N (aq/de)_N}{(aq/d)_N (aq/e)_N} {}_4\phi_3 \begin{bmatrix} aq/bc, d, e, q^{-N}; q, q \\ \deg^{-N}/a, aq/b, aq/c \end{bmatrix}. \] (The formula is incorrectly stated in the paper.) This identity has been applied to prove various partition identities including the Rogers-Ramanujan identities. The author gives a generalization of (**) and several corollaries of the generalization. One of the corollaries furnishes another proof of (*).
Problem 3. Are there multiple series \(q\)-analogs of well-poised hypergeometric series that specialize to either cases of the quintuple product identity, or Winquist’s identity or some other multiple theta series that sums to an infinite product?
L. Winquist’s identity [J. Comb. Theory 6, 56–59 (1969; Zbl 0241.05006)] reads \[ \prod_{n=1}^\infty (1-q^n) (1-zq^n) (1-z^{-1}q^{n-1}) (1-z^2q^{2n-1}) (1-z^{-2}q^{2n-1}) = \sum _{n= -\infty}^\infty q^{(3n^2+n)/2}(z^{3n} - z^{-3-1}); \] it has been generalized by the reviewer and M. V. Subbarao [Proc. Am. Math. Soc. 32, 42–44 (1972; Zbl 0234.05005)].
Problem 4. Are there multiple \(q\)-series analogs of the Chu-Vandermonde summation and Saalschütz’s theorem?
The author defines the \(q\)-analog of the fourth ordinary Lauricella function: \[ \Phi_\beta[a; b_1,\ldots,b_n; c; x_1,\ldots,x_n] = \sum_{m_1,\ldots,m_n\ge 0} \frac{(a)_{m_1+\ldots+m_n} (b_1)_{m_1}\cdots (b_n)_{m_n}x_1^{m_1}\cdots x_n^{m_n}} {(q)_{m_1}\cdots (q)_{m_n}(c)_{m_1+\ldots+m_n}} \]
and proves [J. Lond. Math. Soc. (2) 4, 618–622 (1972; Zbl 0235.33003)] that \[ \Phi_\beta[a; b_1,\ldots,b_n; c; x_1,\ldots,x_n] = \] \[ \frac{(a)_\infty (b_1x_1)_\infty \cdots (b_nx_n)_\infty} {(c)_\infty (x_1)_\infty \cdots (x_n)_\infty} {}_{n+1}\phi_n\begin{bmatrix} c/a, x_1,\ldots, x_n; q,a \\ b_1x_1,\ldots, b_nx_n \qquad \end{bmatrix} . \]
From this is deduced an analog of the Chu-Vandermonde theorem: \[ \Phi_\beta [a; q^{-N+1}/x_1, x_1/x_2, \ldots, x_{n-2}/x_{n-1}, x_{n-1}/q; c; x_1,\ldots, x_{n-1,q}] = \frac{a^N(c/a)_N} {(c)_N}. \]
A special case leads to the result of W. A. Al-Salam [J. Lond. Math. Soc. 40, 455–458 (1965; Zbl 0135.28204)]: \[ \sum_{r,s\ge 0} \frac{(q^{-n})_r (q^{-n})_s (\alpha)_{r+s} (\beta)_r (\beta')_s} {(q)_r (q)_s (\gamma)_{r+s} (\delta)_r (\delta')_s} = \frac{(\beta\beta'/\alpha)_{n+n} (\beta)_n (\beta')_n} {(\beta\beta')_{n+n} (\beta/\alpha)_n (\beta'/\alpha)_n} \]
provided \(\beta'\delta = q^{1-n}\alpha\), \(\beta\delta' = q^{1-n}\alpha\), \(\gamma = \beta\beta'\). Another corollary is an analog of Gauß’s summation involving the Lauricella function \(F_\beta\).
Problem 5. Are there \(q\)-analogs of MacMahon’s Master Theorem and the Dyson Conjecture?
The Master Theorem:
The coefficient of \(x_1^{P_1}\cdots x_n^{P_n}\) in \(\prod_{i=1}^n (A_{i1}x_1 + \ldots + A_{in}x_n)^{P_i}\) is equal to the corresponding coefficient in \((\det(\delta_{ij} - A_{ij}x_j))^{-1}\), where \(\delta_{ij}\) is the Kronecker delta.
Dyson’s Conjecture (proved by J. Gunson [J. Math. Phys. 3, 752–753 (1962; Zbl 0111.43903)], K. G. Wilson [ibid. 3, 1040–1043 (1962; Zbl 0113.21403)] and I. J. Good [ibid. 11, 1884 (1970)] :
The constant term in the expansion of \(\displaystyle\prod_{1\le i\ne j\le n} (1-x_i/x_j)^{a_i}\) is equal to \((a_1+ \ldots +a_n )!/(a_1! \cdots a_n!)\).
Conjecture. The constant term in the expansion of \(\displaystyle\prod_{1\le i\ne j\le n} (x_i\varepsilon_{ij}/x_j)_{a_i}\) is equal to \((q)_{a_1+\ldots+a_n}/(q)_{a_1}\cdots (q)_{a_n})\), where \(\varepsilon_{ij} =1\) if \(i < j\) and \(\varepsilon_{ij} = q\) if \(i> j\). For \(n = 3\) the conjecture reduces to a result of F. H. Jackson [Q. J. Math., Oxf. Ser. 12, 167–172 (1941; Zbl 0063.03007)].
In the remainder of the paper, the author discusses Saalschützian series and inversion theorems.
The paper closes with a bibliography of 63 items.
For the entire collection see [Zbl 0316.00014].

MSC:

33D15 Basic hypergeometric functions in one variable, \({}_r\phi_s\)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
05A30 \(q\)-calculus and related topics
11P83 Partitions; congruences and congruential restrictions
11P84 Partition identities; identities of Rogers-Ramanujan type