Let Φ ( x ) = 1 2 π ∫ − ∞ x     e − 1 2 t 2 d t be the standard normal cumulative distribution function (cdf) and φ ( x ) = 1 2 π e − 1 2 x 2 the standard normal probability density function (pdf). The bivariate standard normal cdf

Φ ρ 2 ( x , y ) = 1 2 π 1 − ρ 2 ∫ − ∞ x     ∫ − ∞ y     e − s 2 − 2 ρ s t + t 2 2 ( 1 − ρ 2 ) d s d t (1)

with a correlation coefficient ρ ∈ ( − 1,1 ) has the bivariate standard normal pdf

φ ρ 2 ( x , y ) = ∂ 2 Φ ρ 2 ( x , y ) ∂ x ∂ y = 1 2 π 1 − ρ 2 e − x 2 − 2 ρ x y + y 2 2 ( 1 − ρ 2 ) . (2)

If ρ = 0 , then Φ 0 2 ( x , y ) = Φ ( x ) Φ ( y ) , while for | ρ | = 1 we have the degenerated cases

Φ ± 1 2 ( x , y ) = ( min { Φ ( x ) , Φ ( y ) } ,   if     ρ = + 1 , max { Φ ( x ) + Φ ( y ) − 1 , 0 } ,   if     ρ = − 1. (3)

Additionally, for ρ ∈ ( − 1,1 ) define

L ( x , y , ρ ) = 1 2 π 1 − ρ 2 ∫ x ∞     ∫ y ∞     e − s 2 − 2 ρ s t + t 2 2 ( 1 − ρ 2 ) d s d t ,

L ( x , y , − 1 ) = max { 1 − Φ ( x ) − Φ ( y ) ,0 } and L ( x , y ,1 ) = 1 − max { Φ ( x ) , Φ ( y ) } . The functions Φ ρ 2 ( x , y ) and L ( x , y , ρ ) are symmetric regarding x and y, and because they are complementary, it suffices to analyze only one of them. In this respect, L is often preferred, but not in this paper where some equations for L are also rewritten using Φ ρ 2 ( x , y ) = L ( − x , − y , ρ ) and some other auxiliary equations.

Noting that (3) implies Φ 1 2 ( x , x ) = Φ ( x ) = 2   Φ 0 2 ( x , 0 ) and Φ − 1 2 ( x , − x ) = 0 = Φ 0 2 ( x , 0 ) + Φ 0 2 ( − x , 0 ) − 1 2 , the equations from ( [1] , p. 937, 26.3.19 and 26.3.20), which do not cover these cases if x ≠ 0 , as we get an indefinite expression 0 0 for the auxiliary correlation coefficients ρ x and ρ y , can be transformed into

Φ ρ 2 ( x , y ) = ( 1 4 + 1 2 π arcsin ρ ,   if     x = y = 0 , Φ ρ x 2 ( x , 0 ) + Φ ρ y 2 ( y , 0 ) − β ,   if     x ≠ 0     or     y ≠ 0 , (4)

where β = 0 if x y > 0 or x y = 0 and x + y ≥ 0 , and β = 1 2 otherwise,

ρ x = { 0 if     | ρ | = 1     and     x ≠ 0     and     x − y sgn ( ρ ) = 0                                         0 ( ρ x − y ) sgn ( x ) x 2 − 2 ρ x y + y 2                                               otherwise ( ρ y − x ) sgn ( y ) x 2 − 2 ρ x y + y 2 } = ρ y , (5)

where sgn ( x ) = 1 , if x ≥ 0 and sgn ( x ) = − 1 , if x < 0 . Note that although | ρ | is not close to 1, | ρ x | or | ρ y | can be, and that | ρ | = 1 and the lower Equations (5) imply | ρ x | = 1 and | ρ y | = 1 .

Since ( [1] , p. 936, 26.3.9 and 26.3.8) imply

Φ ρ 2 ( − x , 0 ) = Φ ρ 2 ( x , 0 ) − Φ ( x ) + 1 2     and     Φ − ρ 2 ( x , 0 ) = Φ ( x ) − Φ ρ 2 ( x , 0 ) , (6)

and Φ − ρ 2 ( − x ,0 ) = 1 2 − Φ ρ 2 ( x ,0 ) , the problem how to compute Φ ρ 2 ( x , y ) is reduced to the computation of Φ ρ 2 ( x ,0 ) for x ≥ 0 and ρ ∈ [ 0,1 ) .

Extensive older bibliography exists about how Φ ρ 2 ( x , y ) can be computed (see Gupta [2] ). Many old methods, e.g. Donnelly’s method [3] based on Owen’s results [4] , Divgi’s method [5] and the method of Drezner and Wesolowsky [6] , are sufficiently effective and accurate for the majority of present needs, but new methods still appear, e.g. [7] [8] [9] [10] . More details and comparisons of some methods can be found in [11] [12] [13] .

The Owen’s T function is defined by ( [4] , pp. 1078-1079)

T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x = arctan a 2 π − 1 2 π ∑ k = 0 ∞ ( − 1 ) k a 2 k + 1 2 k + 1 ( 1 − e − 1 2 h 2 ∑ i = 0 k h 2 i 2 i i ! ) (7)

for a , h ∈ ℝ . It appears frequently in the extensive Owen’s table of integrals involving Gaussian functions ( [14] , pp. 393-413) and can be used for computing Φ ρ 2 ( x , y ) , as will be explained in the next section. The series (7) is used in Donnelly’s algorithm [3] and, based on it, is also realized in pbinorm() function from the VGAM package [15] . Since it alternates, there are problems with numerical stability, e.g. using double precision arithmetic, it may provide only single precision results. More about Owen’s T function and older ways of calculating it are given by Patefield and Tandy [7] , who developed a hybrid method to compute it. To achieve double precision accuracy, they use the series (7) or one of five other methods. Which method is chosen and which its parameters are used depends on which of the 18 regions ( h , a ) belongs to—see ( [7] , p. 7, Fig. 2). Their method is also implemented in OwenT() function from the OwenQ package [16] , hereafter OwenQ::OwenT().

The method presented in this paper extends the theoretical knowledge and can be easily implemented. Its great advantage over the already mentioned methods, which enable Φ ρ 2 ( x , y ) computation via Owen’s T function, is its simplicity, faster convergence and ensured numerical stability. Regarding other methods that have proven to be good or even excellent in terms of Φ ρ 2 ( x , y ) computation speed and accuracy, e.g. the one implemented in the pmvnorm() function from the mvtnorm package [17] , the main advantages are simplicity and independence from various constants. Both enable easy portability between different environments and using high-precision arithmetic. There is no need to adjust the number and precision of various constants, such as weights and knots in quadrature formulas, which are used in many Φ ρ 2 ( x , y ) computational methods.

In Section 2, the background results are presented, which are the starting point for the development of a new method, as well as for the acceleration of the tetrachoric series, which was used for the benchmark values computation. In Section 3, auxiliary results are derived. In core Section 4, new series for Φ ρ 2 ( x ,0 ) are derived in Theorems 3 and 4, and in Corollary 2 they are combined in a unified series for Φ ρ 2 ( x , y ) . However, in all cases, the series must be supplemented by the standard normal cdf and some of them also by the arctangent function computation. Four series for computing the Owen’s T function are presented in Theorem 5. Among them (40b) can be viewed as a modified Euler’s arctangent series. In Sections 5 and 6, algorithmic and numeric issues are discussed, and testing results are presented.

Using double precision arithmetic and testing on a random sample of one million triplets ( x , y , ρ ) with uniformly distributed components, the maximum absolute error was 3.45 × 10⁻¹⁶, which is approximately 3.11-times the machine epsilon 2⁻⁵³. On the same random sample, but with ρ transformed so that more than half of | ρ | were greater than 0.9999, and by equivalent computing of two function values instead of Φ ρ 2 ( x , y ) in rare cases with φ ρ 2 ( x , y ) > 1 , the maximum absolute error was even slightly smaller.

All new series are suitable for vector and high-precision computation. The

fastest one asymptotically converges as ∑ k q k + 1 e − q ( k + 1 ) ! min { a 2 k ,1 } k + 1 ( 1 + a 2 ) k , where q = 1 2 ( 1 + a 2 ) h 2 . Using the Rmpfr package [18] and 1024-bit precision computation, 199 iterations were needed to compute Φ 2 / 2 2 ( 2.1,0 ) , achieving an absolute error smaller than 2 − 1024 ≈ 5.56 × 10 − 309 .

The bivariate standard normal cdf can be expanded into tetrachoric series ( [19] , p. 196), ( [1] , p. 940, 26.3.29)

Φ ρ 2 ( x , y ) = Φ ( x ) Φ ( y ) + φ ( x ) φ ( y ) ∑ k = 0 ∞     H e k ( x ) H e k ( y ) ρ k + 1 ( k + 1 ) ! , (8)

where H e k ( x ) are the “probabilist’s” Hermite polynomials defined by

H e k ( x ) = ( − 1 ) k e 1 2 x 2 d k d x k e − 1 2 x 2   ( k ∈ ℕ 0 ) .

The tetrachoric series (8) converges a bit faster than a geometric series with a quotient ρ and slowly converges for | ρ | close to or equal to 1.

Let ℕ − 1 = ℕ ∪ { − 1,0 } . Extending the usual double factorial function definition for k ∈ ℕ 0 to k ∈ ℕ − 1 with ( − 1 ) ! ! = 1 , we have

H e k ( 0 ) = ( 0,   if   k   is   odd   , ( − 1 ) k 2 ( k − 1 ) ! ! ,   if   k   is   even   .

Setting y = 0 in (8), transforming the summation index k 2 ↦ k , and using ( 2 k − 1 ) ! ! = ( 2 k ) ! 2 k k ! for k ∈ ℕ 0 , we get a slightly faster converging series

Φ ρ 2 ( x ,0 ) = 1 2 Φ ( x ) + φ ( x ) 2 π ∑ k = 0 ∞     H e 2 k ( x ) ( − 1 ) k ρ 2 k + 1 2 k k ! ( 2 k + 1 ) . (9)

Vasicek ( [20] , p. 7) found an alternative series for the computation of Φ ρ 2 ( x ,0 ) , which is a good alternative to the series (9) if ρ 2 > 1 2 .

Integrating the well known equation ( [21] , p. 352)

∂ ∂ ρ Φ ρ 2 ( x , y ) = ∂ 2 Φ ρ 2 ( x , y ) ∂ x ∂ y = φ ρ 2 ( x , y ) (10)

with respect to ρ , using equations Φ 0 2 ( x , y ) = Φ ( x ) Φ ( y ) and (3), we get

Φ ρ 2 ( x , y ) = Φ ( x ) Φ ( y ) + ∫ 0 ρ     φ s 2 ( x , y ) d s = min { Φ ( x ) , Φ ( y ) } − ∫ ρ 1     φ s 2 ( x , y ) d s = max { Φ ( x ) + Φ ( y ) − 1,0 } + ∫ − 1 ρ     φ s 2 ( x , y ) d s . (11)

Due to historical reasons, the variates h and k are often used instead of x and y. We will use h instead of x but will not need k in the role of y.

In the sequel, let ρ ∈ ( − 1,1 ) and r ∈ ℝ be permanently connected by the

one-to-one correspondence r = ρ 1 − ρ 2 and ρ = r 1 + r 2 . Notice that (4) and arcsin ρ = arctan r imply Φ ρ 2 ( 0,0 ) = 1 4 + 1 2 π arctan r . Setting x = h , y = 0 , and writing e − 1 2 h 2 ∫ a b     e 1 2 h 2 φ s 2 ( h ,0 ) d s instead of ∫ a b     φ s 2 ( h ,0 ) d s , the Equations (11) can be rewritten to

Φ ρ 2 ( h ,0 ) = 1 2 Φ ( h ) + φ ( h ) T r ( − 1 2 h 2 ) = min { Φ ( h ) , 1 2 } − φ ( h ) T ˜ r ( − 1 2 h 2 ) = max { Φ ( h ) − 1 2 ,0 } + φ ( h ) T ¯ r ( − 1 2 h 2 ) , (12)

where, using a substitution s = x 1 + x 2 ,

T r ( w ) = 1 2 π ∫ 0 ρ 1 1 − s 2 e w s 2 1 − s 2 d s = 1 2 π ∫ 0 r e w x 2 1 + x 2 d x   ( w ∈ ℝ ) .

Analogously, T ˜ r ( w ) and T ¯ r ( w ) are defined for w ≤ 0 by integrals with limits ( ρ ,1 ) and ( − 1, ρ ) , and ( r , ∞ ) and ( − ∞ , r ) respectively, which do not exist for w > 0 . The computation of Φ ρ 2 ( h ,0 ) is reduced to the computation

of one of the integrals T r ( − 1 2 h 2 ) , T ˜ r ( − 1 2 h 2 ) and T ¯ r ( − 1 2 h 2 ) .

The special cases (12) of the Equations (11) in combination with (4) are a starting point for some computation methods for Φ ρ 2 ( x , y ) which are based on

the integration. Owen’s is one of them—note that φ ( h ) T a ( − 1 2 h 2 ) = T ( h , a ) . Using r instead of a in the sequel and resolving the problem with 0 0 as before, Owen’s equation ( [14] , p. 416, 3.1) can be rewritten as

Φ ρ 2 ( x , y ) = ( 1 4 + 1 2 π arcsin ρ ,   if     x = y = 0 , 1 2 ( Φ ( x ) + Φ ( y ) ) − T ( x , r x ) − T ( y , r y ) − β ,   if     x ≠ 0     or     y ≠ 0 , (13)

where β = 0 if x y > 0 or x y = 0 and x + y ≥ 0 , and β = 1 2 otherwise, and

r x = { 0   if     | ρ | = 1     and     x ≠ 0     and     x − y sgn ( ρ ) = 0                       0 y − ρ x x 1 − ρ 2                                               otherwise   x − ρ y y 1 − ρ 2 } = r y . (14)

Another Owen’s important equation is ( [14] , p. 414, 2.7)

T ( h , r ) + T ( r h , 1 r ) = 1 2 ( Φ ( h ) + Φ ( r h ) ) − Φ ( h ) Φ ( r h ) − β , (15)

where β = 0 if r ≥ 0 and β = 1 2 if r < 0 . It implies φ ( h ) T 1 ( − 1 2 h 2 ) = T ( h ,1 ) = 1 2 Φ ( h ) ( 1 − Φ ( h ) ) . If ρ = 2 2 , then r = 1 . Using (12) and the right Equation (6), we get

Φ 2 / 2 2 ( h , 0 ) = Φ ( h ) ( 1 − 1 2 Φ ( h ) )     and     Φ − 2 / 2 2 ( h , 0 ) = 1 2 ( Φ ( h ) ) 2 . (16)

Let ρ ∈ ( − 1,1 ) , ρ ¯ = sgn ( ρ ) 1 − ρ 2 , r = ρ 1 − ρ 2 , r ¯ = ρ ¯ 1 − ρ ¯ 2 = 1 r and h ∈ ℝ . By adding 1 2 ( Φ ( h ) + Φ ( r h ) ) on both sides of (15) and using (7) and (12), it follows

Φ ρ 2 ( h ,0 ) + Φ ρ ¯ 2 ( r h ,0 ) = Φ ( h ) + Φ ( r h ) − Φ ( h ) Φ ( r h ) − β , (17)

where β = 0 if r ≥ 0 and β = 1 2 if r < 0 .

If | ρ | > 2 2 , then | r | > 1 , | r ¯ | < 1 and | ρ ¯ | < 2 2 . Φ ρ 2 ( h ,0 ) can be computed by computing Φ ρ ¯ 2 ( r h ,0 ) and using (17). Consequently, the right Equation (6) and the transformation of the parameters r h ↦ h ¯ and 1 r ↦ r ¯ , if needed, reduce the condition ρ ∈ ( − 1,1 ) to ρ ∈ [ 0, 2 2 ] with a corresponding r ∈ [ 0,1 ] . The problem how to compute Φ ρ 2 ( x , y ) is reduced to the computation of Φ ρ 2 ( h ,0 ) for h ≥ 0 and ρ ∈ [ 0, 2 2 ] .

The theorems in the sequel do not depend on the restriction to ρ 2 ≤ 1 2 with a corresponding | r | ≤ 1 found in [7] . However, if | r | > 1 , the transformation of r h ↦ h ¯ and 1 r ↦ r ¯ speeds up convergence significantly. This is, for example,

also the case for the Vasicek’s series and the tetrachoric series (9), which transforms into

Φ ρ 2 ( h ,0 ) = Φ ( h ) + 1 2 Φ ( r h ) − Φ ( h ) Φ ( r h )     − φ ( r h ) 2 π ∑ k = 0 ∞     H e 2 k ( r h ) ( − 1 ) k ρ ¯ 2 k + 1 2 k k ! ( 2 k + 1 ) − β , (18)

where ρ ¯ = sgn ( ρ ) 1 − ρ 2 , β = 0 if r ≥ 0 and β = 1 2 if r < 0 . Since the

results calculated with the Vasicek’s series in Section 6 only supplement benchmark values calculated with the tetrachoric series, details about it are not given here.

This paper pays attention to the estimation of the method error due to truncation of infinite series, while the analysis of rounding errors is out of its scope. The following lemma is intended for this. In it and all the theorems below, the phrase “truncated after the n^th term” also means omitting the entire series. In these cases, set n = − 1 and assume the empty sum is 0.

Lemma 1. If the Euler’s series

arctan r = r 1 + r 2 ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! ( r 2 1 + r 2 ) k   ( r ∈ ℝ ) (19)

is truncated after the n^th term, then the remainder

R n = r 1 + r 2 ∑ k = n + 1 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! ( r 2 1 + r 2 ) k   ( n ∈ ℕ − 1 )

has the same sign as r and is bounded in absolute value by

B n ≤ | R n | ≤ ( 1 + r 2 ) B n ,     where     B n = ( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! | r | 1 + r 2 ( r 2 1 + r 2 ) n + 1 . (20)

Proof. Let α ∈ ( 0,1 ] . Noting that k ≥ n + 1 ≥ 0 and ∫ 0 1 ( 1 − x 2 ) k d x = ∫ 0 π 2 cos 2 k + 1 t   d t = ( 2 k ) ! ! ( 2 k + 1 ) ! ! for k ∈ ℕ 0 ( [22] , p. 397, 3.621, 4.), and using Tonelli’s theorem, it follows

S ( r , α ) = ∑ k = n + 1 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! ( α r 2 1 + r 2 ) k = ∑ k = n + 1 ∞ ( α r 2 1 + r 2 ) k ∫ 0 1 ( 1 − x 2 ) k d x = ∫ 0 1 ∑ k = n + 1 ∞ ( α   r 2 ( 1 − x 2 ) 1 + r 2 ) k d x = ( 1 + r 2 ) ( α r 2 1 + r 2 ) n + 1 ∫ 0 1 ( 1 − x 2 ) n + 1 d x 1 + ( 1 − α ) r 2 + α r 2 x 2 . (21)

The denominator in the last integrand is an increasing function of x. Inserting 1 and 0 instead of x implies

( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! α | r | 1 + r 2 ( α r 2 1 + r 2 ) n + 1 ≤ | α r 2 1 + r 2 S ( r , α ) | ≤ ( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! α | r | 1 + ( 1 − α ) r 2 ( α r 2 1 + r 2 ) n + 1 (22)

and, together with B n = r 1 + r 2 S ( r ,1 ) , (20). □

If r ≠ 0 , arctan r = sgn ( r ) π 2 − arctan 1 r , (19) with 1 r instead of r, and r = sgn ( r ) | r | imply

arctan r = sgn ( r ) ( π 2 − | r | 1 + r 2 ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! 1 ( 1 + r 2 ) k )

and (20) with B n = ( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! | r | ( 1 + r 2 ) n + 2 . This series for | r | > 1 converges faster than (19), so we can already sense some kind of similarity with (15).

In the sequel, the lower and upper regularized gamma functions

P ( a , x ) = γ ( a , x ) Γ ( a ) and Q ( a , x ) = Γ ( a , x ) Γ ( a ) respectively, where

γ ( a , x ) = ∫ 0 x     t a − 1 e − t d t and Γ ( a , x ) = ∫ x ∞     t a − 1 e − t d t are the lower and upper incomplete gamma functions respectively, both defined for a > 0 and x ≥ 0 , will be very important, as well as their lower and upper bounds.

In the Alzer’s inequality

( 1 − e − s k + 1 x ) a ≤ P ( a , x ) ≤ ( 1 − e − x ) a   ( a ≥ 1, x ≥ 0 ) , (23)

where s k + 1 = ( ( k + 1 ) ! ) − 1 k + 1 ( [23] , p. 772, Theorem 1), ( [24] , p. 221, (5.4)), allowing a = 1 resulted in strict inequalities becoming non-strict. It implies

1 − ( 1 − e − x ) a ≤ Q ( a , x ) ≤ 1 − ( 1 − e − s k + 1 x ) a . (24)

For n ∈ ℕ 0 and x ≥ 0 , we have I n ( x ) = e x P ( n + 1, x ) ( [22] , p. 908, 8.352, 1.), where I n ( x ) = e x − ∑ k = 0 n x k k ! is the remainder of the Taylor series of the exponential function. The Lagrangian form of the remainder I n ( x ) = x n + 1 e ξ n ( x ) ( n + 1 ) ! , where 0 < ξ n ( x ) < x , implies I n ( x ) ≥ x n + 1 ( n + 1 ) ! for x ≥ 0 . Since I n ( x ) < n + 2 n + 2 − x x n + 1 ( n + 1 ) ! for x ∈ ( 0, n + 2 ) ( [25] , pp. 323-324, 3.8.25), it follows

x k + 1 e − x ( k + 1 ) ! ≤ P ( k + 1, x ) ≤ 2 x k + 1 e − x ( k + 1 ) !   ( k ∈ ℕ 0 ,0 ≤ x ≤ k + 2 2 ) , (25)

where the equality holds only for x = 0 .

Theorem 2. Let f ^ ( x ) = ∑ k = 0 ∞     c ^ k x 2 k and f ˜ ( x ) = ∑ k = 0 ∞     c ˜ k x 2 k + 1 be absolutely convergent series for x ∈ ℝ . Then

F ^ ( x ) = ∫ 0 x     φ ( β t ) f ^ ( t ) d t = 1 2 ∑ k = 0 ∞ ( 2 k − 1 ) ! ! c ^ k | β | 2 k + 1 P ( k + 1 2 , 1 2 β 2 x 2 ) (26)

and

F ˜ ( x ) = ∫ 0 x     φ ( β t ) f ˜ ( t ) d t = 1 2 π ∑ k = 0 ∞ ( 2 k ) ! ! c ˜ k β 2 k + 2 P ( k + 1, 1 2 β 2 x 2 ) (27)

for x ≥ 0 and β ≠ 0 .

Proof. Let f k ( t ) = c ^ k φ ( β t ) t 2 k and F k ( x ) = ∫ 0 x     f k ( t ) d t for x ≥ 0 . Using ( 2 k − 1 ) ! ! = 2 k π Γ ( k + 1 2 ) and a substitution 1 2 β 2 t 2 = u , we get

F k ( x ) = c ^ k 2 π ∫ 0 x     e − 1 2 β 2 t 2 t 2 k   d t = 2 k − 1 c ^ k π | β | 2 k + 1 ∫ 0 1 2 β 2 x 2     u k − 1 2 e − u   d u = ( 2 k − 1 ) ! ! c ^ k 2 | β | 2 k + 1 γ ( k + 1 2 , 1 2 β 2 x 2 ) Γ ( k + 1 2 ) = ( 2 k − 1 ) ! ! c ^ k 2 | β | 2 k + 1 P ( k + 1 2 , 1 2 β 2 x 2 ) .

Setting j = ⌈ β 2 x 2 ⌉ + 1 implies 0 ≤ u < j + 2 2 for u ∈ [ 0, 1 2 β 2 x 2 ] . Then

∑ k = 0 ∞ | F k ( x ) | = ∑ k = 0 j | F k ( x ) | + ∑ k = j + 1 ∞ | F k ( x ) | , where the first sum is finite. Using the fact that P ( a , x ) is a decreasing function of a ( [26] , p. 154), (25) and the absolute convergence assumption of f ^ ( x ) , it follows

P ( k + 1 2 , 1 2 β 2 x 2 ) ≤ P ( k , 1 2 β 2 x 2 ) ≤ 2 β 2 k x 2 k   e − 1 2 β 2 x 2 2 k k !   ( k ≥ j + 1 )

and

∑ k = j + 1 ∞ | F k ( x ) | = ∑ k = j + 1 ∞ ( 2 k − 1 ) ! ! | c ^ k | 2 | β | 2 k + 1 P ( k + 1 2 , 1 2 β 2 x 2 ) ≤ e − 1 2 β 2 x 2 | β | ∑ k = j + 1 ∞ ( 2 k − 1 ) ! ! ( 2 k ) ! ! | c ^ k | x 2 k < ∞ .

Noting that ∫ 0 x | f k ( t ) | d t = | F k ( x ) | , we proved that ∑ k = 0 ∞ ∫ 0 x | f k ( t ) | d t < ∞ . By Fubini’s theorem it follows that ∫ 0 x     φ ( β t ) f ^ ( t ) d t = ∑ k = 0 ∞     F k ( x ) , hence (26).

Let f k ( t ) = c ˜ k φ ( β t ) t 2 k + 1 and F k ( x ) = ∫ 0 x     f k ( t ) d t for x ≥ 0 . Analogously to above, and using ( 2 k ) ! ! = 2 k k ! = 2 k Γ ( k + 1 ) , we get F k ( x ) = ( 2 k ) ! ! c ˜ k 2 π   β 2 k + 2 P ( k + 1, 1 2 β 2 x 2 ) ,

∑ k = j + 1 ∞ | F k ( x ) | ≤ | x | e − 1 2 β 2 x 2 2 π ∑ k = j + 1 ∞ | c ˜ k | | x | 2 k + 1 k + 1 < ∞

and (27). □

Note that in the same way as the Equations (26) and (27) were derived, analogous equations can be derived if the integration limits 0 and x are replaced by x and ∞ respectively, and lower regularized gamma function is replaced by the upper one. However, for each case it should be checked whether the order of integration and summation can be reversed.

The usefulness of Theorem 2 is that if the sequences { c ^ k } and { c ˜ k } can be computed recursively, then the terms of the series (26) and (27) can be too. Let

q = 1 2 β 2 x 2 . Crucial is the calculation of d ^ k = Q ( k + 1 2 , q ) and d ˜ k = P ( k + 1, q ) ,

k ∈ ℕ 0 , the rest is trivial. Since P ( k + 1, q ) = 1 − Q ( k + 1, q ) , we will calculate d k = Q ( k + 1, q ) instead of d ˜ k = 1 − d k .

Using P ( 1 2 , q ) = 2 Φ ( 2 q ) − 1 ( [1] , p. 934, 26.2.30), we get d ^ 0 = 1 − P ( 1 2 , q ) = 2 ( 1 − Φ ( 2 q ) ) , and d 0 = e − q . Since P ( a + 1, x ) = P ( a , x ) − x a   e − x Γ ( a + 1 ) ( [1] , p. 262, 6.5.21) implies Q ( a + 1, x ) = Q ( a , x ) + x a   e − x Γ ( a + 1 ) , using auxiliary variables b ^ k = q k − 1 2   e − q Γ ( k + 1 2 ) and b k = q k   e − q Γ ( k + 1 ) , it follows

b ^ 0 = e − q q π ,   d ^ 0 = 2 ( 1 − Φ ( 2 q ) )   and   b ^ k + 1 = q b ^ k k + 1 2 ,   d ^ k + 1 = d ^ k + b ^ k + 1

for q ≠ 0 and k = 0,1,2,... , and

b 0 = e − q ,   d 0 = b 0   and   b k + 1 = q b k k + 1 ,   d k + 1 = d k + b k + 1 (28)

for k = 0,1,2,...

The core of this article are Theorems 3, 4 and 5. Since the first two do not specify variants for the transformed parameters, the last one, together with (13) and (14), is more practically useful. It also justifies the title of the article.

Theorem 3. Let h ∈ ℝ , ρ ∈ ( − 1,1 ) , r = ρ 1 − ρ 2 and q = 1 2 ( 1 + r 2 ) h 2 = h 2 2 ( 1 − ρ 2 ) . Then

Φ ρ 2 ( h ,0 ) = 1 2 Φ ( h ) + arctan r 2 π − r 2 π ( 1 + r 2 ) ∑ k = 0 ∞     c k ( r 2 1 + r 2 ) k = 1 2 Φ ( h ) + arcsin ρ 2 π − 1 − ρ 2 2 π ∑ k = 0 ∞     c k ρ 2 k + 1 , (29)

where c k = ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, q ) . The series S = ∑ k = 0 ∞     c k ( r 2 1 + r 2 ) k converges for every r , h ∈ ℝ . If it is truncated after the n^th term, then

Φ ρ 2 ( h ,0 ) = 1 2 Φ ( h ) + arctan r 2 π − r S n 2 π ( 1 + r 2 ) − sgn ( r ) R n   ( n ∈ ℕ − 1 ) , (30)

where S n = ∑ k = 0 n     c k ( r 2 1 + r 2 ) k and R n = | r | 2 π ( 1 + r 2 ) ∑ k = n + 1 ∞     c k ( r 2 1 + r 2 ) k is bounded by

0 ≤ R n ≤ B n 1 + e − q r 2 ≤ B n , (31)

where B n = ( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! | r | 2 π ( 1 − e − q ) n + 2 ( r 2 1 + r 2 ) n + 1 .

Proof. From (7) and Φ ( x ) − 1 2 = φ ( x ) ∑ k = 0 ∞ x 2 k + 1 ( 2 k + 1 ) ! ! for x ∈ ℝ ( [1] , p. 932, 26.2.11), it follows

∂ T ( h , r ) ∂ h = − h 2 π ∫ 0 r     e − 1 2 h 2 ( 1 + x 2 )   d x = − φ ( h ) ( Φ ( r h ) − 1 2 ) = φ ( β h ) ∑ k = 0 ∞     c ˜ k h 2 k + 1 , (32)

r , h ∈ ℝ , where β = 1 + r 2 and c ˜ k = − r 2 k + 1 2 π ( 2 k + 1 ) ! ! . Since

∑ k = 0 ∞     c ˜ k h 2 k + 1 = − φ ( h ) φ ( β h ) ( Φ ( r h ) − 1 2 ) = − e 1 2 r 2 h 2 ( Φ ( r h ) − 1 2 ) ,

the series on the left side converges for every r , h ∈ ℝ and is an odd function of r h . If r h < 0 , then all its terms are positive, implying that it converges absolutely.

All requirements of Theorem 2 are fulfilled. Assuming r ≥ 0 and using T ( 0, r ) = arctan r 2 π , (32) and (27), it follows

T ( h , r ) = arctan r 2 π − 1 2 π ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! r 2 k + 1 β 2 k + 2 P ( k + 1, 1 2 β 2 h 2 ) ,

which, together with (12), (7), r 2 β 2 = r 2 1 + r 2 = ρ 2 , r 1 + r 2 = ρ 1 − ρ 2 and 1 2 β 2 h 2 = q , imply (29) for r ≥ 0 . Since (7) implies T ( h , − r ) = − T ( h , r ) , (29) is also valid for r < 0 . Let α = 1 − e − q . Using (23), (21) and (22), it follows

0 ≤ R n ≤ α | r | 2 π ( 1 + r 2 ) ∑ k = n + 1 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! ( α r 2 1 + r 2 ) k = α | r | 2 π ( 1 + r 2 ) S ( r , α )

and (31). □

Note that if r h ≠ 0 , then c k > 0 and c k + 1 c k = 2 k + 2 2 k + 3 P ( k + 2 , q ) P ( k + 1 , q ) < 1 . Hence the sequence { c k ( r 2 1 + r 2 ) k } is decreasing.

Corollary 1. For every h ∈ ℝ we have

Φ ( h ) = 1 2 + sgn ( h ) S ,     where     S = 1 2 π ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, h 2 ) 2 k . (33)

Proof. Using (29) with ρ = − 2 2 , hence r = − 1 , q = h 2 and the right Equation (16), it follows

1 2 Φ ( h ) − 1 8 + 1 4 π ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, h 2 ) 2 k = 1 2 ( Φ ( h ) ) 2

and a quadratic equation ( Φ ( h ) ) 2 − Φ ( h ) + 1 4 − S = 0 . If h = 0 , then S = 0 and (33) is valid. If h ≠ 0 , since Φ ( h ) > 1 2 if h > 0 and Φ ( h ) < 1 2 if h < 0 , only 1 2 + sgn ( h ) S is a proper root of the two. Note that sending h → ∞ implies ∑ k = 0 ∞ k ! ( 2 k + 1 ) ! ! = π 2 . □

Theorem 4. Let h ∈ ℝ , ρ ∈ ( − 1,1 ) , r = ρ 1 − ρ 2 and q = 1 2 ( 1 + r 2 ) h 2 = h 2 2 ( 1 − ρ 2 ) . Then

Φ ρ 2 ( h ,0 ) = 1 2 Φ ( h ) + r 2 π ( 1 + r 2 ) ∑ k = 0 ∞     c k ( r 2 1 + r 2 ) k = 1 2 Φ ( h ) + 1 − ρ 2 2 π ∑ k = 0 ∞     c k ρ 2 k + 1 , (34)

where c k = ( 2 k ) ! ! ( 2 k + 1 ) ! ! Q ( k + 1, q ) . The series S = ∑ k = 0 ∞     c k ( r 2 1 + r 2 ) k converges for every r , h ∈ ℝ . If it is truncated after the n^th term, then

Φ ρ 2 ( h ,0 ) = 1 2 Φ ( h ) + r S n 2 π ( 1 + r 2 ) + sgn ( r ) R n   ( n ∈ ℕ − 1 ) , (35)

where S n = ∑ k = 0 n     c k ( r 2 1 + r 2 ) k and R n = | r | 2 π ( 1 + r 2 ) ∑ k = n + 1 ∞     c k ( r 2 1 + r 2 ) k is bounded by

0 ≤ R n ≤ B n ,     where     B n = ( 2 n + 2 ) ! ! ( 2 n + 3 ) ! ! | r | 2 π ( r 2 1 + r 2 ) n + 1 . (36)

Proof. Using (29), (19), P ( k + 1, x ) = 1 − Q ( k + 1, x ) , r 2 1 + r 2 = ρ 2 and

r 1 + r 2 = ρ 1 − ρ 2 , it follows (34). Noting that R n = | r | 2 π ( 1 + r 2 ) S ( r ,1 ) and (22) imply (36). □

Note that the upper bound (24) is not suitable for a simple estimation of R n here because lim k → ∞ s k + 1 = 0 , hence Q ( k + 1, q ) ≤ 1 was implicitly used instead.

Q ( a , x ) is an increasing function of a. The ratio c k + 1 c k = 2 k + 2 2 k + 3 Q ( k + 2, q ) Q ( k + 1, q ) can be greater than 1, e.g. if h = 3 and r = 1 , we get c 2 c 1 = 4 Q ( 3 , 9 ) 5 Q ( 2 , 9 ) ≈ 4.04 . In Theorem 4, the sequence { c k ( r 2 1 + r 2 ) k } is not always decreasing, but, in any case, the well-behaved arctan | r | series (19) divided by 2π is a majorant series for | r | S 2 π ( 1 + r 2 ) from Theorems 3 and 4.

Corollary 2. Let x , y ∈ ℝ , x 2 + y 2 > 0 , ρ ∈ ( − 1,1 ) , ρ x = ( ρ x − y ) sgn ( x ) x 2 − 2 ρ x y + y 2 , ρ y = ( ρ y − x ) sgn ( y ) x 2 − 2 ρ x y + y 2 and q = x 2 − 2 ρ x y + y 2 2 ( 1 − ρ 2 ) . Let c k = ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, q ) and c ˜ k = ( 2 k ) ! ! ( 2 k + 1 ) ! ! Q ( k + 1, q ) for k ∈ ℕ 0 . Then

Φ ρ 2 ( x , y ) = 1 2 ( Φ ( x ) + Φ ( y ) ) + arcsin ρ x + arcsin ρ y 2 π − S − β = 1 2 ( Φ ( x ) + Φ ( y ) ) + S ˜ − β , (37)

where

S = 1 2 π 2 q ∑ k = 0 ∞     c k ( | x | ρ x 2 k + 1 + | y | ρ y 2 k + 1 ) ,

S ˜ is S with c ˜ k instead of c k , and β = 0 if x y > 0 or x y = 0 and x + y ≥ 0 , and β = 1 2 otherwise.

Proof. Since q x = x 2 2 ( 1 − ρ x 2 ) = x 2 − 2 ρ x y + y 2 2 ( 1 − ρ 2 ) = q and q y = y 2 2 ( 1 − ρ y 2 ) = x 2 − 2 ρ x y + y 2 2 ( 1 − ρ 2 ) = q , 1 − ρ x 2 = | x | 1 − ρ 2 x 2 − 2 ρ x y + y 2 = | x | 2 q and

1 − ρ y 2 = | y | 2 q , (4), (5), (29) and (34) imply (37). □

Note that r 1 + r 2 = r ¯ 1 + r ¯ 2 and q = 1 2 ( 1 + r 2 ) h 2 = 1 2 ( 1 + r ¯ 2 ) h ¯ 2 = q ¯ , therefore they are invariant on the transformation of r h ↦ h ¯ and 1 r ↦ r ¯ , but S n , R n and B n from Theorems 3 and 4 are not because ρ 2 = r 2 1 + r 2 ↦ 1 1 + r 2 = r ¯ 2 1 + r ¯ 2 = ρ ¯ 2 and ρ 2 ≠ ρ ¯ 2 if | r | ≠ 1 . They transform to S ¯ n , R ¯ n and B ¯ n . By transforming the parameters and using (17), the Equations (30) and (35) can be rewritten to

Φ ρ 2 ( h , 0 ) = 1 2 Φ ( r h ) + Φ ( h ) ( 1 − Φ ( r h ) ) − arctan r ¯ 2 π + r S ¯ n 2 π ( 1 + r 2 ) + sgn ( r ) R ¯ n − β (38)

and

Φ ρ 2 ( h , 0 ) = 1 2 Φ ( r h ) + Φ ( h ) ( 1 − Φ ( r h ) ) − r S ¯ n 2 π ( 1 + r 2 ) − sgn ( r ) R ¯ n − β (39)

respectively, where β = 0 if r ≥ 0 and β = 1 2 if r < 0 .

The transformation makes sense if | r | > 1 , implying ρ ¯ 2 < 1 2 and faster con

vergence. For example, the inequality 0 ≤ R n ≤ B n from Theorem 4 transforms into 0 ≤ R ¯ n ≤ B ¯ n , where the upper bound B ¯ n is r 2 n + 4 -times smaller than a corresponding upper bound B n . The noteworthy cost for faster convergence is the additional computation of Φ ( r h ) .

Theorem 5. If | r | < ∞ , the Owen’s T function (7) can be written as

T ( h , r ) = arctan r 2 π − r 2 π ( 1 + r 2 ) ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, q ) ( r 2 1 + r 2 ) k (40a)

= r 2 π ( 1 + r 2 ) ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! Q ( k + 1, q ) ( r 2 1 + r 2 ) k (40b)

= U ( h , r ) − arctan r 2 π + r 2 π ( 1 + r 2 ) ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! P ( k + 1, q ) ( 1 1 + r 2 ) k (40c)

= U ( h , r ) − r 2 π ( 1 + r 2 ) ∑ k = 0 ∞ ( 2 k ) ! ! ( 2 k + 1 ) ! ! Q ( k + 1, q ) ( 1 1 + r 2 ) k , (40d)

where q = 1 2 ( 1 + r 2 ) h 2 , U ( h , r ) = 1 2 ( Φ ( h ) + Φ ( r h ) ) − Φ ( h ) Φ ( r h ) − β , and β = 0 if r ≥ 0 and β = 1 2 if r < 0 .

Proof. Using (12) and (7), the Equations (29) and (34) imply (40a) and (40b) respectively, which, together with (17), imply (40c) and (40d) respectively. □

Note that when truncating the series (40a) and (40b), the truncation error in absolute value can be estimated by (31) and (36) respectively. The same upper

bounds with r ¯ = 1 r instead of r also apply to (40c) and (40d).

Because of (40b) and 0 ≤ Q ( k + 1, q ) ≤ 1 , the Owen’s T function can be seen as a modified Euler’s arctangent series (19) divided by 2π, which is its majorant series.

Let S ˜ = r S 2 π ( 1 + r 2 ) and e ˜ k = B k , where S and B k relate to Theorem 4. Using (34) and (28), the series S ˜ can be calculated by Recursion 1 (tildes are omitted). Analogously, let S ^ = r S 2 π ( 1 + r 2 ) and e ^ k = B k , where S and B k relate to Theorem

3. The series S ^ can be calculated by Recursion 2 (hats are omitted). In practice, both recursions can be easily combined, since only four steps, marked with ⊳ , are different.

In both cases, given ε > 0 , the recursion is terminated when the condition e k + 1 < ε is met and S k + 1 is taken as its result. However, if ε > 0 is too small, or even intentionally negative, the calculation proceeds up to the actual accuracy threshold—the recursion is terminated when the condition S k + 1 ≤ S k is met. This criterion was used in the calculations presented in the next section.

In both recursions, all variables are positive and are bounded upward. If q ≠ 0 , the sequence { d k } strictly increases and, due to (24), converges to 1. The sequence { S k } is strictly increasing if r ≠ 0 . If r = 0 , the recursion terminates

with S 1 = 0 . Note that (7) implies | T ( r , h ) | ≤ arctan | r | 2 π . If step 7 in Recursion 2 is replaced by S 0 ← a 0 ( 1 − d 0 ) − arctan | r | 2 π , the sequence { S k } changes its sign but keeps the direction. However, the modified Recursion 2 computes S ^ = r S 2 π ( 1 + r 2 ) − arctan r 2 π .

Let S ˜ n and S ^ n be the results of Recursion 1 and modified Recursion 2 respectively, calculated with the same h and r if | r | ≤ 1 , or with h ¯ and r ¯ if | r | > 1 . Let Φ ˜ ρ 2 ( h ,0 ) be calculated by (35) or (39), where unknown remainders are ignored. Analogously, let Φ ^ ρ 2 ( h ,0 ) be calculated by (30) or (38), where also

arctan r 2 π is ignored. If | r | ≤ 1 , then Φ ˜ ρ 2 ( h ,0 ) ≤ Φ ρ 2 ( h ,0 ) ≤ Φ ^ ρ 2 ( h ,0 ) if r ≥ 0

and Φ ˜ ρ 2 ( h ,0 ) ≥ Φ ρ 2 ( h ,0 ) ≥ Φ ^ ρ 2 ( h ,0 ) if r < 0 . Even if | r | > 1 , one of Φ ˜ ρ 2 ( h ,0 ) and Φ ^ ρ 2 ( h ,0 ) underestimates and the other overestimates Φ ρ 2 ( h ,0 ) . Consequently, the upper bounds (31) and (36) are not needed if S ˜ and modified S ^ are calculated in the same recursion which terminates when | S ^ k + 1 − S ˜ k + 1 | become small enough. Upper bounds are also not needed if in Recursions 1 and 2 step 17 is replaced so that S k + 1 ≤ S k is the only termination condition. In this case, steps 3, 8 and 15 are unnecessary.

Given q, which one of the series (29) and (34) converges faster? Since P ( k + 1, x ) is a cdf of the gamma distribution with the median

ν ∈ ( k + 2 3 , k + 1 ) , it follows P ( k + 1, q ) ≤ Q ( k + 1, q ) if q ≤ ν < k + 1 . The an

swer depends on the number of considered terms, which depends on the prescribed accuracy. On this basis, it seems that series (29) is more suitable than (34) for “small” q and vice versa for “large” q. Table 1 and Table 4 in the next section show how this is reflected in practice.

Gautschi’s inequality x 1 − s < Γ ( x + 1 ) Γ ( x + s ) < ( x + 1 ) 1 − s ( [27] , p. 138, 5.6.4) with s = 1 2 implies k π 2 k + 1 < ( 2 k ) ! ! ( 2 k + 1 ) ! ! < ( k + 1 ) π 2 k + 1 . Combining it with (25), sending k → ∞ and neglecting non-essential constants imply that the series (29) asymptotically converges as ∑ k q k + 1 e − q ( k + 1 ) ! ρ 2 k k + 1 . Assuming calculation with h ¯ , r ¯ and (38) instead of h, r and (30) respectively if | r | > 1 , hence ρ 2 > 1 2 , and recalling that q is invariant on the transformation of r h ↦ h ¯ and 1 r ↦ r ¯ , we get an asymptotic convergence as ∑ k q ¯ k + 1 e − q ¯ ( k + 1 ) ! ρ ¯ 2 k k + 1 = ∑ k q k + 1 e − q ( k + 1 ) ! ( 1 − ρ 2 ) k k + 1 . The series (29) or its corresponding transformed version, unwritten and based on (40c), asymptotically converges as ∑ k q k + 1 e − q ( k + 1 ) ! ( min { ρ 2 , 1 − ρ 2 } ) k k + 1 = ∑ k q k + 1 e − q ( k + 1 ) ! min { r 2 k , 1 } k + 1 ( 1 + r 2 ) k , where q = 1 2 ( 1 + r 2 ) h 2 = h 2 2 ( 1 − ρ 2 ) . On the other hand, since (24) implies lim k → ∞ Q ( k + 1, x ) = 1 , the series (34) or its corresponding transformed version, unwritten and based on (40d), asymptotically converges as ∑ k ( min { ρ 2 ,1 − ρ 2 } ) k k + 1 = ∑ k min { r 2 k ,1 } k + 1 ( 1 + r 2 ) k , hence in the worst case as ∑ k 1 2 k k + 1 .

Based on both asymptotic convergences, comparing (34) to (29) would be unfair because the calculation of arctan | r | should also be taken into account for (29). Since it is implicitly embedded in Recursion 1, it is calculated on the fly at the cost of worse speed of convergence. Each of the recursions is executable with the basic four arithmetic operations, two calls of elementary functions (if the calculation of r from ρ is also considered), and arctan | r | in the modified Recursion 2. However, the computational cost of Φ ( h ) (and Φ ( r h ) if needed) should also be taken in account, but it is the same in (29) and (34).

A common cause of numerical instability, where two absolutely large values give an absolutely small result when added or subtracted, cannot occur in Recursions 1 and 2. The calculation by them is stable, but something may happen that needs

to be pointed out. The side effect of transforming r h ↦ h ¯ and 1 r ↦ r ¯ is that

| h ¯ | can become unusually large. When computing Φ ( h ) , according to Marsaglia ( [28] , p. 2), “[...] the region of interest may not be the statistician’s 0 < x < 5 , say, but rather 10 < x < 14 [...]’’. Assume the same needs when computing Φ ρ 2 ( h ,0 ) ,

and using transformed parameters if ρ 2 > 1 2 . If h = 14 and

ρ = 10 1 + 10 2 ≈ 0.995 , then r ≈ 10 , r ¯ ≈ 0.1 and h ¯ ≈ 140 , implying that the

region of interest could be | h | < 140 . Even the usual | h | < 5 transformed to | h ¯ | < 50 is well beyond the established limits.

The “problem” of too large | h | is mentioned because we have to be aware of it if we are going to perform high-precision computation. In any case, the recursion must be complemented by a suitable computation of Φ ( h ) and arctan r if needed. As an example that not all are like that, the computation by the well

known series Φ ( x ) = 1 2 + 1 2 π ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k k ! ( 2 k + 1 ) ( [1] , p. 932, 26.2.10) and the

standard double precision arithmetic is numerically unstable even for moderately large | h | , e.g. using the programming language R [29] , for h = 10 we get a completely wrong result, approximately −1683. A similar problem can lead to a significant decrease in accuracy if T ( h , r ) is calculated with (7), as will be seen in Subsection 6.1.

In double precision arithmetic, the minimal positive non-zero number is η ≈ 2.23 × 10 − 308 . Assuming b 0 = e − q ≥ η , it follows q max = − log ( η ) ≈ 708.40 and

h max ( r ) = − 2 log ( η ) 1 + r 2 . If | r | ≤ 1 , then h max ( r ) is between h max ( 1 ) ≈ 26.62 and

h max ( 0 ) ≈ 37.64 . If e − q < η , then all elements of the sequence { b k } are zero. Consequently, all elements of { d k } are one and all elements of { S k } in Recursion 2 are zero.

From the integral in (7), it follows | T ( h , r ) | ≤ φ ( h ) Φ ( | r | | h | ) − 1 2 | h | ≤ φ ( h ) 2 | h | if h ≠ 0 , where the upper bound is a decreasing function of | h | . If e − q < η and | r | ≤ 1 , then | h | > h max ( 1 ) and | T ( h , r ) | ≤ φ ( h max ( 1 ) ) 2   h max ( 1 ) ≈ 1.12 × 10 − 156 . Analogously to the case when r = 0 , the recursion terminates with S 1 = 0 , which is practically a correct result for T ( h , r ) , as well as Φ ρ 2 ( h , 0 ) = 1 2 if h > 0 , and Φ ρ 2 ( h , 0 ) = 0 if h < 0 .

The Recursions 1 and 2 with adjustment of their results according to (40a)—(40d) are implemented in the package Phi2rho [30] . It contains the functions OwenT(), hereafter Phi2rho::OwenT(), and Phi2xy(), which compute T ( h , r ) and Φ ρ 2 ( x , y ) respectively. They also enable computation with the tetrachoric and Vasicek’s series, original or accelerated. The latter means that h and r are transformed if r > 1 and r < 1 for the tetrachoric and Vasicek’s series respectively, and the Equation (15) is used, so (18) instead of (9) in the case of tetrachoric series. For the sake of comparison, all series were considered during testing in Subsection 6.1. In Subsection 6.2, the original tetrachoric and Vasicek’s series were excluded because they converge too slowly for unfavorable ρ .

During testing, the values of T ( r , h ) and Φ ρ 2 ( x , y ) were computed for a selected and randomly generated set of parameters. Reference values were computed using the accelerated tetrachoric series and quadruple (128-bit) precision. Absolute differences between tested and reference values are declared as absolute errors. The maximum absolute error is considered a measure of accuracy. Since the mean values and some quantiles are also important, they are shown in the tables in Subsection 6.2. An integral part of the testing was also a comparison of the results obtained with the Phi2rho package and competing packages available on CRAN.

If the generally applicable Theorem 2 is not considered, Theorems 3 and 4, and Corollary 2 are interesting mainly theoretically, because Theorem 5, together with (13) and (14), replaces and supplements them in practice. The series (40a)—(40d) enable fast and numerically stable computation, which is often more important than the speed of convergence. From Table 1 and Table 4 can be concluded that the computation with (40a) or (40c) needs significantly fewer iterations than the computation with the accelerated tetrachoric and Vasicek’s series, and from Table 5 and Table 6 can be assumed that the advantage increases with increasing precision. For 53-bit precision (double precision), it required a half, and for 1024-bit precision only a fifth, of the iterations demanded by the competing series. However, the cost of calculating arctan r is not taken into account. In connection with this, let us just mention two more questions.

Recalling that e − x I n ( x ) = P ( n + 1, x ) , the Taylor series of the arctangent function and (7) imply

T ( h , r ) = arctan r 2 π − 1 2 π ∑ k = 0 ∞ ( − 1 ) k r 2 k + 1 2 k + 1 P ( k + 1, 1 2 h 2 ) = 1 2 π ∑ k = 0 ∞ ( − 1 ) k r 2 k + 1 2 k + 1 Q ( k + 1, 1 2 h 2 ) ,

which Fayed and Atiya ( [32] , p. 241) have already noticed. If the above series, (29) and (34) are viewed as functions of the variable h, they have a similar structure, but the coefficients, depending on r, belong to different arctangent series. Whether the similarity could be explained by a similar Euler transform as the one which transforms the Taylor arctangent series to the Euler’s arctangent series seems interesting question, but was not deeply explored.

In Recursion 1, the external calculation of the arctangent function is avoided, assuming that the Euler’s series (19) is used for it, and the expectation that there would be no difference in the number of iterations for Recursions 1 and 2 if those for calculating the arctangent were also taken into account. However, the use of the external arctangent calculation allows the use of already known and possible future faster converging series. Such is the case with

arctan r = r 1 + r 2 ∑ k = 0 ∞ ( 2 k − 1 ) ! ! ( 2 k ) ! ! ( 2 k + 1 ) ( r 2 1 + r 2 ) k   ( r ∈ ℝ ) , (42)

which is based on the Taylor series of arcsin r and arctan r = arcsin r 1 + r 2 .

Indeed, the number of iterations is not significantly smaller if | r | ≤ 1 and the accuracy obtained by double precision computation is sufficient, and even the square root must be calculated, but it could be significantly smaller in high-precision computation. The Euler’s series (19) is included in [1] and [27] , but (42) is not, even though it converges faster. It can be found in ( [22] , p. 61, 1.644 1.) as

arctan r = r 1 + r 2 ∑ k = 0 ∞ ( 2 k ) ! 2 2 k ( k ! ) 2 ( 2 k + 1 ) ( r 2 1 + r 2 ) k

with a reference to the 1922 source. Due to equations ( 2 k − 1 ) ! ! = ( 2 k ) ! 2 k k ! and

( 2 k ) ! ! = 2 k k ! , the coefficients are the same as those in (42). Whether we could also find a corresponding series for the Owen’s T function, which would converge faster than (40b), remains an open challenge.

The author declares no conflicts of interest regarding the publication of this paper.

Komelj, J. (2023) The Bivariate Normal Integral via Owen’s T Function as a Modified Euler’s Arctangent Series. American Journal of Computational Mathematics, 13, 476-504. https://doi.org/10.4236/ajcm.2023.134026