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The supported rising factorials are given in the following table (note that only a few are supported in `Q`): +------------+-------------------+--------+ | Key | Rising factorial | `Q` | +============+===================+========+ | ``'m'`` | `(a_j)_m` | Yes | +------------+-------------------+--------+ | ``'n'`` | `(a_j)_n` | Yes | +------------+-------------------+--------+ | ``'m+n'`` | `(a_j)_{m+n}` | Yes | +------------+-------------------+--------+ | ``'m-n'`` | `(a_j)_{m-n}` | No | +------------+-------------------+--------+ | ``'n-m'`` | `(a_j)_{n-m}` | No | +------------+-------------------+--------+ | ``'2m+n'`` | `(a_j)_{2m+n}` | No | +------------+-------------------+--------+ | ``'2m-n'`` | `(a_j)_{2m-n}` | No | +------------+-------------------+--------+ | ``'2n-m'`` | `(a_j)_{2n-m}` | No | +------------+-------------------+--------+ For example, the Appell F1 and F4 functions .. math :: F_1 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b)_m (c)_n}{(d)_{m+n}} \frac{x^m y^n}{m! n!} F_4 = \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} \frac{(a)_{m+n} (b)_{m+n}}{(c)_m (d)_{n}} \frac{x^m y^n}{m! n!} can be represented respectively as ``hyper2d({'m+n':[a], 'm':[b], 'n':[c]}, {'m+n':[d]}, x, y)`` ``hyper2d({'m+n':[a,b]}, {'m':[c], 'n':[d]}, x, y)`` More generally, :func:`~mpmath.hyper2d` can evaluate any of the 34 distinct convergent second-order (generalized Gaussian) hypergeometric series enumerated by Horn, as well as the Kampe de Feriet function. The series is computed by rewriting it so that the inner series (i.e. the series containing `n` and `y`) has the form of an ordinary generalized hypergeometric series and thereby can be evaluated efficiently using :func:`~mpmath.hyper`. If possible, manually swapping `x` and `y` and the corresponding parameters can sometimes give better results. **Examples** Two separable cases: a product of two geometric series, and a product of two Gaussian hypergeometric functions:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> x, y = mpf(0.25), mpf(0.5) >>> hyper2d({'m':1,'n':1}, {}, x,y) 2.666666666666666666666667 >>> 1/(1-x)/(1-y) 2.666666666666666666666667 >>> hyper2d({'m':[1,2],'n':[3,4]}, {'m':[5],'n':[6]}, x,y) 4.164358531238938319669856 >>> hyp2f1(1,2,5,x)*hyp2f1(3,4,6,y) 4.164358531238938319669856 Some more series that can be done in closed form:: >>> hyper2d({'m':1,'n':1},{'m+n':1},x,y) 2.013417124712514809623881 >>> (exp(x)*x-exp(y)*y)/(x-y) 2.013417124712514809623881 Six of the 34 Horn functions, G1-G3 and H1-H3:: >>> from mpmath import * >>> mp.dps = 10; mp.pretty = True >>> x, y = 0.0625, 0.125 >>> a1,a2,b1,b2,c1,c2,d = 1.1,-1.2,-1.3,-1.4,1.5,-1.6,1.7 >>> hyper2d({'m+n':a1,'n-m':b1,'m-n':b2},{},x,y) # G1 1.139090746 >>> nsum(lambda m,n: rf(a1,m+n)*rf(b1,n-m)*rf(b2,m-n)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) 1.139090746 >>> hyper2d({'m':a1,'n':a2,'n-m':b1,'m-n':b2},{},x,y) # G2 0.9503682696 >>> nsum(lambda m,n: rf(a1,m)*rf(a2,n)*rf(b1,n-m)*rf(b2,m-n)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) 0.9503682696 >>> hyper2d({'2n-m':a1,'2m-n':a2},{},x,y) # G3 1.029372029 >>> nsum(lambda m,n: rf(a1,2*n-m)*rf(a2,2*m-n)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) 1.029372029 >>> hyper2d({'m-n':a1,'m+n':b1,'n':c1},{'m':d},x,y) # H1 -1.605331256 >>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m+n)*rf(c1,n)/rf(d,m)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) -1.605331256 >>> hyper2d({'m-n':a1,'m':b1,'n':[c1,c2]},{'m':d},x,y) # H2 -2.35405404 >>> nsum(lambda m,n: rf(a1,m-n)*rf(b1,m)*rf(c1,n)*rf(c2,n)/rf(d,m)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) -2.35405404 >>> hyper2d({'2m+n':a1,'n':b1},{'m+n':c1},x,y) # H3 0.974479074 >>> nsum(lambda m,n: rf(a1,2*m+n)*rf(b1,n)/rf(c1,m+n)*\ ... x**m*y**n/fac(m)/fac(n), [0,inf], [0,inf]) 0.974479074 **References** 1. [SrivastavaKarlsson]_ 2. [Weisstein]_ http://mathworld.wolfram.com/HornFunction.html 3. 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In order for the series to make sense, none of the parameters may be integers. **Examples** The value of `\,_2H_2` at `z = 1` is given by Dougall's formula:: >>> from mpmath import * >>> mp.dps = 25; mp.pretty = True >>> a,b,c,d = 0.5, 1.5, 2.25, 3.25 >>> bihyper([a,b],[c,d],1) -14.49118026212345786148847 >>> gammaprod([c,d,1-a,1-b,c+d-a-b-1],[c-a,d-a,c-b,d-b]) -14.49118026212345786148847 The regularized function `\,_1H_0` can be expressed as the sum of one `\,_2F_0` function and one `\,_1F_1` function:: >>> a = mpf(0.25) >>> z = mpf(0.75) >>> bihyper([a], [], z) (0.2454393389657273841385582 + 0.2454393389657273841385582j) >>> hyper([a,1],[],z) + (hyper([1],[1-a],-1/z)-1) (0.2454393389657273841385582 + 0.2454393389657273841385582j) >>> hyper([a,1],[],z) + hyper([1],[2-a],-1/z)/z/(a-1) (0.2454393389657273841385582 + 0.2454393389657273841385582j) **References** 1. [Slater]_ (chapter 6: "Bilateral Series", pp. 180-189) 2. 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