Make rules consistent with ChainRules definitions in SpecialFunctions

Author: devmotion

src/rules.jl

@@ -140,7 +140,7 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
 @define_diffrule SpecialFunctions.airybiprime(x) =
     :(  $x * SpecialFunctions.airybi($x)  )
 @define_diffrule SpecialFunctions.airyaix(x)     =
-    :(  sqrt($x) * SpecialFunctions.airyaix($x) + SpecialFunctions.airyaiprimex($x)  )
+    :(  SpecialFunctions.airyaiprimex($x) + sqrt($x) * SpecialFunctions.airyaix($x)  )
 @define_diffrule SpecialFunctions.airyaiprimex(x) =
     :(  $x * SpecialFunctions.airyaix($x) + sqrt($x) * SpecialFunctions.airyaiprimex($x)  )
 @define_diffrule SpecialFunctions.besselj0(x)    =
@@ -171,15 +171,15 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
 @define_diffrule SpecialFunctions.besselk(ν, x)   =
     :NaN, :( -(SpecialFunctions.besselk($ν - 1, $x) + SpecialFunctions.besselk($ν + 1, $x)) / 2  )
 @define_diffrule SpecialFunctions.besselkx(ν, x)  =
-    :NaN, :( -(SpecialFunctions.besselkx($ν - 1, $x) - 2SpecialFunctions.besselkx($ν, $x) + SpecialFunctions.besselkx($ν + 1, $x)) / 2 )
+    :NaN, :( -(SpecialFunctions.besselkx($ν - 1, $x) + SpecialFunctions.besselkx($ν + 1, $x)) / 2 + SpecialFunctions.besselkx($ν, $x)  )
 @define_diffrule SpecialFunctions.hankelh1(ν, x)  =
     :NaN, :(  (SpecialFunctions.hankelh1($ν - 1, $x) - SpecialFunctions.hankelh1($ν + 1, $x)) / 2  )
 @define_diffrule SpecialFunctions.hankelh1x(ν, x) =
-    :NaN, :(  (SpecialFunctions.hankelh1x($ν - 1, $x) - 2im*SpecialFunctions.hankelh1x($ν, $x) - SpecialFunctions.hankelh1x($ν + 1, $x)) / 2  )
+    :NaN, :(  (SpecialFunctions.hankelh1x($ν - 1, $x) - SpecialFunctions.hankelh1x($ν + 1, $x)) / 2 - im * SpecialFunctions.hankelh1x($ν, $x)  )
 @define_diffrule SpecialFunctions.hankelh2(ν, x)  =
     :NaN, :(  (SpecialFunctions.hankelh2($ν - 1, $x) - SpecialFunctions.hankelh2($ν + 1, $x)) / 2  )
 @define_diffrule SpecialFunctions.hankelh2x(ν, x)  =
-    :NaN, :(  (SpecialFunctions.hankelh2x($ν - 1, $x) + 2im*SpecialFunctions.hankelh2x($ν, $x) - SpecialFunctions.hankelh2x($ν + 1, $x)) / 2  )
+    :NaN, :(  (SpecialFunctions.hankelh2x($ν - 1, $x) - SpecialFunctions.hankelh2x($ν + 1, $x)) / 2 + im * SpecialFunctions.hankelh2x($ν, $x)  )
 @define_diffrule SpecialFunctions.gamma(a, x)     = 
     :NaN, :(-exp(-$x) * $x^($a - 1))
 @define_diffrule SpecialFunctions.polygamma(m, x) =