are real valued functions of the parameters and argument, typically involving products of gamma functions. When these are degenerate, finite limiting cases exist. Hence for

x < 0

{}_{2}F_{1} (a, b; c; x)

is defined by analytic continuation, and for

x < 1

{}_{2}F_{1} (a, b; c; x)

is real and finite.

For

x = 1

, the following apply:

If $c > a + b$ , ${}_{2}F_{1} (a, b; c; 1) = \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)}$ , and hence is finite. Solutions also exist for the degenerate cases where $c - a$ or $c - b$ are negative integers or zero.
If $c \leq a + b$ , ${}_{2}F_{1} (a, b; c; 1)$ is infinite, and the sign of ${}_{2}F_{1} (a, b; c; 1)$ is determinable as $x$ approaches $1$ from below.

In the complex plane, the principal branch of

{}_{2}F_{1} (a, b; c; z)

is taken along the real axis from

x = 1.0

increasing.

{}_{2}F_{1} (a, b; c; z)

is multivalued along this branch, and for real parameters

a, b

and

c

is typically not real valued. As such, this function will not compute a solution when

x > 1

The solution strategy used by this function is primarily dependent upon the value of the argument

x

. Once trivial cases and the case

x = 1.0

are eliminated, this proceeds as follows.

For

0 < x \leq 0.5

, sets of safe parameters

{α_{i, j}, β_{i, j}, ζ_{i, j}, χ_{j} | 1 \leq j \leq 2 |, 1 \leq i \leq 4}

are determined, such that the values of

{}_{2}F_{1} (a_{j}, b_{j}; c_{j}; x_{j})

required for an appropriate transformation of the type (3) may be calculated either directly or using recurrence relations from the solutions of

{}_{2}F_{1} (α_{i, j}, β_{i, j}; ζ_{i, j}; χ_{j})

. If

c

is positive, then only transformations with

C_{2} = 0.0

will be used, implying only

{}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1})

will be required, with the transformed argument

x_{1} = x

. If

c

is negative, in some cases a transformation with

C_{2} \neq 0.0

will be used, with the argument

x_{2} = 1.0 - x

. The function then cycles through these sets until acceptable solutions are generated. If no computation produces an accurate answer, the least inaccurate answer is selected to complete the computation. See Section 7.

For

0.5 < x < 1.0

, an identical approach is first used with the argument

x

. Should this fail, a linear transformation resulting in both transformed arguments satisfying

x_{j} = 1.0 - x

is employed, and the above strategy for

0 < x \leq 0.5

is utilized on both components. Further transformations in these sub-computations are however, limited to single terms with no argument transformation.

For

x < 0

, a linear transformation mapping the argument

x

to the interval

(0, 0.5]

is first employed. The strategy for

0 < x \leq 0.5

is then used on each component, including possible further two term transforms. To avoid some degenerate cases, a transform mapping the argument

x

[0.5, 1)

may also be used.

For improved precision in the final result, this function accepts

a, b

and

c

split into an integral and a decimal fractional component. Specifically,

a = a_{i} + a_{r}

, where

| a_{r} | \leq 0.5

and

a_{i} = a - a_{r}

is integral. The other parameters

b

and

c

are similarly deconstructed.

In addition to the above restrictions on

c

and

x

, an artificial bound, arbnd, is placed on the magnitudes of

a, b, c

and

x

to minimize the occurrence of overflow in internal calculations, particularly those involving real to integer conversions.

arbnd = 0.0001 \times I_{\max}

, where

I_{\max}

is the largest machine integer (see X02BBC). It should however, not be assumed that this function will produce accurate answers for all values of

a, b, c

and

x

satisfying this criterion.

This function also tests for non-finite values of the parameters and argument on entry, and assigns non-finite values upon completion if appropriate. See Section 9 and Chapter X07.

Please consult the NIST Digital Library of Mathematical Functions for a detailed discussion of the Gauss hypergeometric function including special cases, transformations, relations and asymptotic approximations.

4 References

NIST Digital Library of Mathematical Functions

Pearson J (2009) Computation of hypergeometric functions MSc Dissertation, Mathematical Institute, University of Oxford

5 Arguments

1: $ani$ – double Input

On entry:

a_{i}

, the nearest integer to

a

, satisfying

a_{i} = a - a_{r}

Constraints:

$ani = ⌊ ani ⌋$ ;
$| ani | \leq arbnd$ .

2: $adr$ – double Input

On entry:

a_{r}

, the signed decimal remainder satisfying

a_{r} = a - a_{i}

and

| a_{r} | \leq 0.5

Constraint:

| adr | \leq 0.5

3: $bni$ – double Input

On entry:

b_{i}

, the nearest integer to

b

, satisfying

b_{i} = b - b_{r}

Constraints:

$bni = ⌊ bni ⌋$ ;
$| bni | \leq arbnd$ .

4: $bdr$ – double Input

On entry:

b_{r}

, the signed decimal remainder satisfying

b_{r} = b - b_{i}

and

| b_{r} | \leq 0.5

Constraint:

| bdr | \leq 0.5

5: $cni$ – double Input

On entry:

c_{i}

, the nearest integer to

c

, satisfying

c_{i} = c - c_{r}

Constraints:

$cni = ⌊ cni ⌋$ ;
$| cni | \leq arbnd$ ;
if $| cdr | < 16.0 ε$ , $cni \geq 1.0$ .

6: $cdr$ – double Input

On entry:

c_{r}

, the signed decimal remainder satisfying

c_{r} = c - c_{i}

and

| c_{r} | \leq 0.5

Constraint:

| cdr | \leq 0.5

7: $x$ – double Input

On entry: the argument

x

Constraint:

- arbnd < x \leq 1

8: $frf$ – double * Output

On exit:

f_{fr}

, the scaled real component of the solution satisfying

f_{fr} = {}_{2}F_{1} (a, b; c; x) \times 2^{- f_{sc}}

, i.e.,

{}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}}

. See Section 9 for the behaviour of

f_{fr}

when a finite or non-finite answer is returned.

9: $scf$ – Integer * Output

On exit:

f_{sc}

, the scaling power of two, satisfying

f_{sc} = \log_{2} (\frac{{}_{2}F_{1} (a, b; c; x)}{f_{fr}})

, i.e.,

{}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}}

. See Section 9 for the behaviour of

f_{sc}

when a non-finite answer is returned.

10: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_CANNOT_CALCULATE: An internal calculation has resulted in an undefined result.
NE_COMPLEX: On entry, $x = ⟨ value ⟩$ .
In general, ${}_{2}F_{1} (a, b; c; x)$ is not real valued when $x > 1$ .
NE_INFINITE: On entry, $x = ⟨ value ⟩$ , $c = ⟨ value ⟩$ , $a + b = ⟨ value ⟩$ .
${}_{2}F_{1} (a, b; c; 1)$ is infinite in the case $c \leq a + b$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_OVERFLOW: Overflow occurred in a subcalculation of ${}_{2}F_{1} (a, b; c; x)$ . The answer may be completely incorrect.
NE_REAL: On entry, adr does not satisfy $| adr | \leq 0.5$ .

On entry, bdr does not satisfy $| bdr | \leq 0.5$ .

On entry, cdr does not satisfy $| cdr | \leq 0.5$ .
NE_REAL_2: On entry, $c = cni + cdr = ⟨ value ⟩$ .
${}_{2}F_{1} (a, b; c; x)$ is undefined when $c$ is zero or a negative integer.
NE_REAL_ARG_NON_INTEGRAL: ani is non-integral.
On entry, $ani = ⟨ value ⟩$ .
Constraint: $ani = ⌊ ani ⌋$ .

bni is non-integral.
On entry, $bni = ⟨ value ⟩$ .
Constraint: $bni = ⌊ bni ⌋$ .

cni is non-integral.
On entry, $cni = ⟨ value ⟩$ .
Constraint: $cni = ⌊ cni ⌋$ .
NE_REAL_RANGE_CONS: On entry, ani does not satisfy $| ani | \leq arbnd = ⟨ value ⟩$ .

On entry, bni does not satisfy $| bni | \leq arbnd = ⟨ value ⟩$ .

On entry, cni does not satisfy $| cni | \leq arbnd = ⟨ value ⟩$ .

On entry, x does not satisfy $| x | \leq arbnd = ⟨ value ⟩$ .
NE_TOTAL_PRECISION_LOSS: All approximations have completed, and the final residual estimate indicates no accuracy can be guaranteed.
$Relative residual = ⟨ value ⟩$ .
NW_OVERFLOW_WARN: On completion, overflow occurred in the evaluation of ${}_{2}F_{1} (a, b; c; x)$ .
NW_SOME_PRECISION_LOSS: All approximations have completed, and the final residual estimate indicates some precision may have been lost.
$Relative residual = ⟨ value ⟩$ .
NW_UNDERFLOW_WARN: Underflow occurred during the evaluation of ${}_{2}F_{1} (a, b; c; x)$ . The returned value may be inaccurate.

7 Accuracy

In general, if

fail . code =

NE_NOERROR, the value of

{}_{2}F_{1} (a, b; c; x)

may be assumed accurate, with the possible loss of one or two decimal places. Assuming the result does not overflow, an error estimate

res

is made internally using equation (1). If the magnitude of this residual

res

is sufficiently large, a different fail.code will be returned. Specifically,

$fail . code =$ NE_NOERROR or NW_UNDERFLOW_WARN	$res \leq 1000 ε$
$fail . code =$ NW_SOME_PRECISION_LOSS	$1000 ε < res \leq 0.1$
$fail . code =$ NE_TOTAL_PRECISION_LOSS	$res > 0.1$

where

ε

is the machine precision as returned by X02AJC. Note that underflow may also have occurred if

fail . code =

NE_TOTAL_PRECISION_LOSS or NW_SOME_PRECISION_LOSS.

A further estimate of the residual can be constructed using equation (1), and the differential identity,

\begin{matrix} \frac{d ({}_{2}F_{1} (a, b; c; x))}{d x} = \frac{a b}{c} {}_{2}F_{1} (a + 1, b + 1; c + 1; x) \\ \frac{d^{2} ({}_{2}F_{1} (a, b; c; x))}{d x^{2}} = \frac{a (a + 1) b (b + 1)}{c (c + 1)} {}_{2}F_{1} (a + 2, b + 2; c + 2; x) \end{matrix}

(4)

This estimate is however, dependent upon the error involved in approximating

{}_{2}F_{1} (a + 1, b + 1; c + 1; x)

and

{}_{2}F_{1} (a + 2, b + 2; c + 2; x)

8 Parallelism and Performance

s22bfc is not threaded in any implementation.

9 Further Comments

s22bfc returns non-finite values when appropriate. See Chapter X07 for more information on the definitions of non-finite values.

Should a non-finite value be returned, this will be indicated in the value of fail, as detailed in the following cases.

fail . code =

NE_NOERROR or

fail . code =

NE_TOTAL_PRECISION_LOSS, NW_SOME_PRECISION_LOSS or NW_UNDERFLOW_WARN, a finite value will have been returned with approximate accuracy as detailed in Section 7.

The values of

f_{f r}

and

f_{s c}

are implementation dependent. In most cases, if

{}_{2}F_{1} (a, b; c; x) = 0

f_{f r} = 0

and

f_{s c} = 0

will be returned, and if

{}_{2}F_{1} (a, b; c; x)

is finite, the fractional component will be bound by

0.5 \leq | f_{f r} | < 1

, with

f_{s c}

chosen accordingly.

The values returned in frf (

f_{fr}

) and scf (

f_{sc}

) may be used to explicitly evaluate

{}_{2}F_{1} (a, b; c; x)

, and may also be used to evaluate products and ratios of multiple values of

{}_{2}F_{1}

as follows,

\begin{matrix} {}_{2}F_{1} (a, b; c; x) = f_{fr} \times 2^{f_{sc}} \\ {}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1}) \times {}_{2}F_{1} (a_{2}, b_{2}; c_{2}; x_{2}) = (f_{fr 1} \times f_{fr 2}) \times 2^{(f_{sc 1} + f_{sc 2})} \\ \frac{{}_{2}F_{1} (a_{1}, b_{1}; c_{1}; x_{1})}{{}_{2}F_{1} (a_{2}, b_{2}; c_{2}; x_{2})} = \frac{f_{fr 1}}{f_{fr 2}} \times 2^{(f_{sc 1} - f_{sc 2})} \\ \ln | {}_{2}F_{1} (a, b; c; x) | = \ln | f_{fr} | + f_{sc} \times \ln (2) . \end{matrix}

fail . code =

NE_INFINITE then

{}_{2}F_{1} (a, b; c; x)

is infinite. A signed infinity will have been returned for frf, and

scf = 0

. The sign of frf should be correct when taking the limit as

x

approaches

1

from below.

fail . code =

NW_OVERFLOW_WARN then upon completion,

| {}_{2}F_{1} (a, b; c; x) | > 2^{I_{\max}}

, where

I_{\max}

is given by X02BBC, and hence is too large to be representable even in the scaled form. The scaled real component returned in frf may still be correct, whilst

scf = I_{\max}

will have been returned.

fail . code =

NE_OVERFLOW then overflow occurred during a subcalculation of

{}_{2}F_{1} (a, b; c; x)

. The same result as for

fail . code =

NW_OVERFLOW_WARN will have been returned, however, there is no guarantee that this is representative of either the magnitude of the scaling power

f_{sc}

, or the scaled component

f_{fr}

{}_{2}F_{1} (a, b; c; x)

fail . code =

NE_NOERROR, frf and scf were inaccessible to s22bfc, and as such it is not possible to determine what their values may be following the call to s22bfc.

For all other error exits,

scf = 0

will be returned and frf will be returned as a signalling NaN (see x07bbc).

fail . code =

NE_CANNOT_CALCULATE an internal computation produced an undefined result. This may occur when two terms overflow with opposite signs, and the result is dependent upon their summation for example.

fail . code =

NE_REAL_2 then

c

is too close to a negative integer or zero on entry, and

{}_{2}F_{1} (a, b; c; x)

is undefined. Note, this will also be the case when

c

is a negative integer, and a (possibly trivial) linear transformation of the form (3) would result in either:

(i)all $c_{j}$ not being negative integers,
(ii)for any $c_{j}$ which remain as negative integers, one of the corresponding parameters $a_{j}$ or $b_{j}$ is a negative integer of magnitude less than $c_{j}$ .

In the first case, the transformation coefficients

C_{j} (a_{j}, b_{j}, c_{j}, x_{j})

are typically either infinite or undefined, preventing a solution being constructed. In the second case, the series (2) will terminate before the degenerate term, resulting in a polynomial of fixed degree, and hence potentially a finite solution.

fail . code =

NE_REAL_RANGE_CONS then no computation will have been performed due to the risk of integer overflow. The actual solution may however, be finite.

fail . code =

NE_COMPLEX indicates

x > 1

, and hence the requested solution is on the boundary of the principal branch of

{}_{2}F_{1} (a, b; c; x)

. Hence it is multivalued, typically with a nonzero imaginary component. It is however, strictly finite.

10 Example

This example evaluates the Gauss hypergeometric function at two points in scaled form using s22bfc, and subsequently calculates their product and ratio implicitly.