parameter differentiation for bessel and associated legendre functions › ~dlozier › sf21 ›...

parameter di�erentiation for Bessel andassociated Legendre functions

Howard S. Cohl?†

? Information Technology Laboratory, National Institute of Standards and Technology,

Gaithersburg, Maryland, U.S.A.

† Department of Mathematics, University of Auckland, Auckland, New Zealand

International Conference on Special Functionsin the 21st Century: Theory and Applications

Washington, D.C., U.S.A.

April 7, 2011

Howard Cohl (NIST) parameter di�erentiation April 7, 2011 1 / 31

Outline

1 introduction � special functions

2 parameter di�erentiation � Bessel functions

3 parameter di�erentiation � associated Legendre functions

4 some particular cases for Legendre functions

5 references

6 summary


introduction � special functions

de�nitions: special functions

gamma function (Re z > 0) Γ(z + 1) = zΓ(z)

Γ(z) :=∫ ∞

0tz−1e−tdt

Euler re�ection formula: Γ(z)Γ(1− z) =π

sinπzfactorial: (for n ∈ {0, 1, 2, . . .}) Γ(n+ 1) = n!

Pochhammer symbol: (for n ∈ {0, 1, 2, . . .}) (a)n :=Γ(a+ n)

Γ(a)digamma function � logarithmic derivative of gamma function

d

dzΓ(z) =: Γ(z)ψ(z)

generalized factorial function

pFq ((ap); (bq); z) =∞∑n=0

(a1)n(a2)n . . . (ap)n(b1)n(b2)n . . . (bq)n

zk

k!



de�nitions � Bessel functions (unrestricted order ν)

Bessel function of the �rst kind

Jν(z) =(z/2)ν

Γ(ν + 1)0F1

(ν + 1;−z

2

4

)=(z

2

)ν ∞∑n=0

(−z2/4)n

n!Γ(ν + n+ 1)

Bessel function of the second kind (Weber's function)

Yν(z) = cot(πν)Jν(z)− csc(πν)J−ν(z)Yn(z) = 1

π∂Jν(z)∂ν

∣∣∣ν=n

+ (−1)n

π∂Jν(z)∂ν

∣∣∣ν=−n

modi�ed Bessel function of the �rst kind

Iν(z) =(z/2)ν

Γ(ν + 1)0F1

(ν + 1;

z2

4

)=(z

2

)ν ∞∑n=0

(z2/4)n

n!Γ(ν + n+ 1)

modi�ed Bessel function of the second kind (Macdonald's function)

Kν(z) = π2 csc(πν)I−ν(z)− csc(πν)Iν(z)

Kn(z) = (−1)n

2∂I−ν(z)∂ν

∣∣∣ν=n− (−1)n

2∂Iν(z)∂ν

∣∣∣ν=n



associated Legendre functions P µν , Q

µν : C \ {−1, 1} → C

(complex degree ν order µ)

two (Gauss hypergeometric) functions � associated Legendre d.e.

(1− z2)d2w

dz2− 2z

dw

dz+[ν(ν + 1)− µ2

1− z2

]w = 0

associated Legendre function of the �rst kind (|1− z| < 2)

Pµν (z) = 1Γ(1−µ)

[z+1z−1

]µ2

2F1

(−ν, ν + 1; 1− µ; 1−z

2

)= − sin(πν)

π

(z+1z−1

)µ2∞∑n=0

Γ(−ν + n)Γ(ν + 1 + n)n!Γ(1− µ+ n)

(1− z

2

)nassociated Legendre function of the second kind (|z| > 1)

Qµν (z) =√πeiµπΓ(ν+µ+1)(z2−1)µ/2

2ν+1Γ(ν+ 32)zν+µ+1 2F1

(ν+µ+2

2 , ν+µ+12 ; ν + 3

2 ; 1z2

)= 1

2eiπµ(z2 − 1)

µ2

∞∑n=0

Γ(ν+µ+22 + n)Γ(ν+µ+1

2 + n)n!Γ(ν + 3

2 + n)z2n



order derivative illustration for modi�ed Bessel functions


parameter di�erentiation � Bessel functions

order derivatives for Bessel functions


∂J±ν(z)∂ν

= ±J±ν(z) log(z

2

)∓∞∑m=0

(−1)m(z

2

)±ν+2m ψ(±ν +m+ 1)m!Γ(±ν +m+ 1)


∂Yν(z)∂ν

= cot(πν)∂Jν(z)∂ν

− csc(πν)∂J−ν(z)∂ν

− π csc(πν)Y−ν(z)


∂Iν(z)∂ν

= Iν(z) log(z

2

)−∞∑m=0

(z2

)ν+2m ψ(ν +m+ 1)m!Γ(ν +m+ 1)


∂Kν(z)∂ν

= −π cot(πν)Kν(z) +π

2csc(πν)

[∂I−ν(z)∂ν

− ∂Iν(z)∂ν

]Howard Cohl (NIST) parameter di�erentiation April 7, 2011 7 / 31


Bessel function mirror symmetry about ν = 0


J−ν(z) = Jν(z) cos(πν)− Yν(z) sin(πν)


Y−ν(z) = Jν(z) sin(πν) + Yν(z) cos(πν)


I−ν(z) = Iν(z) +2πKν(z) sin(πν)


K−ν(z) = Kν(z)



order derivatives for Bessel functions (zero-order)

Derivatives of Bessel function with respect to the order evaluated at

integer-orders is given in �3.2.3 in Magnus, Oberhettinger & Soni (1966)

Bessel function of the �rst kind[∂

∂νJν(z)

]ν=0

=π

2Y0(z)

Bessel function of the second kind (Weber's function)[∂

∂νYν(z)

]ν=0

= −π2J0(z)

modi�ed Bessel function of the �rst kind[∂

∂νIν(z)

]ν=0

= −K0(z)

modi�ed Bessel function of the second kind (Macdonald's function)[∂

∂νKν(z)

]ν=0

= 0



order derivatives for Bessel functions (integer-order)


integer-orders is given by (�3.2.3 in MOS (1966))


∂νJν(z)

]ν=±n

=π

2(±1)nYn(z)± (±1)n

n!2

n−1∑k=0

(z/2)k−n

k!(n− k)Jk(z)

Bessel function of the second kind (Weber's function)[∂

∂νYν(z)

]ν=±n

= −π2

(±1)nJn(z)± (±1)nn!2

n−1∑k=0

(z/2)k−n

k!(n− k)Yk(z)

modi�ed Bessel function of the �rst kind[∂

∂νIν(z)

]ν=±n

= (−1)n+1Kn(z)± n!2

n−1∑k=0

(−z/2)k−n

k!(n− k)Ik(z)

modi�ed Bessel function of the second kind (Macdonald's function)[∂

∂νKν(z)

]ν=±n

= ±n!2

n−1∑k=0

(z/2)k−n

k!(n− k)Kk(z)



order derivatives for Bessel functions (half-integer)


half-integer-orders is given by (�3.3.3 in MOS (1966))


∂νJν(z)

]ν=±1/2

=

√2πz

[{sin zcos z

}Ci(2z)∓

{cos zsin z

}Si(2z)

]Bessel function of the second kind (Weber's function)[∂

∂νYν(z)

]ν=±1/2

= ±√

2πz

[{cos zsin z

}Ci(2z)±

{sin zcos z

}[Si(2z)− π]

]modi�ed Bessel function of the �rst kind[∂

∂νIν(z)

]ν=±1/2

=

√1

2πz[ezEi(−2z)∓ e−zEi(2z)

]modi�ed Bessel function of the second kind (Macdonald's function)[∂

∂νKν(z)

]ν=±1/2

= ∓√

π

2zezEi(−2z)



order derivatives for Bessel functions

cosine integral: Ci(z) := −∫ ∞z

cos ttdt

sine integral: Si(z) :=∫ z

0

sin ttdt

exponential integral: Ei(z) := −∫ ∞−z

e−t

tdt

order derivatives at half-odd integer orders[∂

∂νJν(z)

]ν=1/2+n

= ci(2z)Jn+1/2(z)− (−1)nSi(2z)J−n−1/2(z)

+n!2

n−1∑k=0

(z/2)k−n

k!(n− k)Jk+1/2(z)− n!

√πz

2

n∑k=1

(2/z)k

(n− k)!k

k−1∑p=0

zp

p!

×[Jn−k+1/2(z)Jp−1/2(2z)− (−1)n−k−pJk−n−1/2(z)J1/2−p(2z)

]Howard Cohl (NIST) parameter di�erentiation April 7, 2011 12 / 31

parameter di�erentiation � associated Legendre functions

parameter di�erentiation for associated Legendre functions

w1 = Pµν : {z ∈ C : |z − 1| < 2} → C

Pµν (z) = −sin(πν)π

(z + 1z − 1

)µ2∞∑n=0

Γ(−ν+n)Γ(ν+1+n)n!Γ(1−µ+n)

(1− z

2

)nQµν (z) = − πiπµΓ(1+ν+µ)

2 sin(πµ)Γ(1+ν−µ)P−µν (z) + πeiπµ

2 sin(πµ)Pµν (z)

w2 = Qµν : {z ∈ C : |z| > 1} → C

Qµν (z) = 12eiπµ(z2 − 1)

µ2

∞∑n=0

Γ(ν+µ+22 + n)Γ(ν+µ+1

2 + n)n!Γ(ν + 3

2 + n)z2n

Pµν (z) = e−iπµ sin[π(ν+µ)]π cos(πν) Qµν (z)− e−iπµ sin[π(ν−µ)]

π cos(πν) Qµ−ν−1(z)

would like∂Pµν (z)∂ν

,∂Qµν (z)∂ν

, and∂Pµν (z)∂µ

,∂Qµν (z)∂µ

, for z ∈ C \{−1,1}



domains in C: P µν (z) : |z − 1| < 2 Qµ

ν(z) : |z| > 1

-2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3



general degree derivatives for associated Legendre functions

general degree derivatives for {z ∈ C : |z − 1| < 2}∂∂νP

µν (z) = π cotπνPµν (z)− 1

πAµν (z)

∂∂νQ

µν (z) = π cot(πν)Qµν (z)

− πeiπµΓ(1+ν+µ)2 sin(πµ)Γ(1+ν−µ) [ψ(1 + ν − µ)− ψ(1 + ν − µ)]P−µν (z)

+ eiπµΓ(1+ν+µ)2 sin(πµ)Γ(1+ν−µ)A

−µν (z)− eiπµ

2 sin(πµ)Aµν (z)

∂∂νQ

µν (z) = −π2

2 Pµν (z) + π sin(πµ)

sin(πν) sinπ(ν+µ)Qµν (z)− 1

2 cotπ(ν + µ)Aµν (z)

+12 cscπ(ν + µ)Aµ

ν (−z)

Aµν (z) = sin(πν)

(z+1z−1

)µ2∞∑n=0

Γ(−ν+n)Γ(ν+1+n)n!Γ(1−µ+n)

[ψ(ν+n+1)−ψ(−ν+n)](

1−z2

)nPµν (−z) = e∓iπνPµν (z)− 2

πe−iπµ sin[π(ν + µ)]Qµν (z)

Qµν (−z) = −e±iπνQµν (z)



general order derivatives for associated Legendre functions

general order derivatives for {z ∈ C : |z − 1| < 2}∂

∂µPµν (z) =

12Pµν (z) log

z + 1z − 1

− 1πBµν (z)

∂

∂µQµν (z) = − πΓ(1+ν+µ)eiπµ

2Γ(1+ν−µ) sin(πµ) [ψ(1 + ν + µ) + ψ(1 + ν − µ)]P−µν (z)

+π[i− cot(πµ)]Qµν (z) + 12 log z+1

z−1Qµν (z)

+ Γ(1+ν+µ)eiπµ

Γ(1+ν−µ) sin(πµ)B−µν (z)− eiπµ

2 sin(πµ)Bµν (z)

Bµν (z) = sin(πν)

(z + 1z − 1

)µ2∞∑n=0

Γ(ν + n+ 1)ψ(n− µ+ 1)n!Γ(ν − n+ 1)Γ(n− µ+ 1)

(1− z

2

)n



general order derivatives for associated Legendre functions

general order derivatives for {z ∈ C : |z| > 1}∂

∂µQµν (z) = [iπ +

12

log(z2 − 1)]Qµν (z) +14Cµν (z)

∂

∂µPµν (z) = [iπ + 1

2 log(z2 − 1)]Pµν (z)

+ e−2iπµ

cos(πν)

[e−iπνQµν (z) + eiπνQµ−ν−1(z)

]+ e−iπν

4π cos(πν)

{sin[π(ν + µ)]Cµ

ν (z)− sin[π(ν − µ)]Cµ−ν−1(z)

}Cµν (z) = eiπµ(z2 − 1)

µ2

×∞∑n=0

Γ( ν+µ+22

+n)Γ( ν+µ+12

+n)n!Γ(ν+ 3

2+n)z2n

[ψ(ν+µ+2

2 + n) + ψ(ν+µ+12 + n)

]



general degree derivatives for associated Legendre functions

general degree derivatives for {z ∈ C : |z| > 1}∂

∂νQµν (z) =

14Cµν (z)− 1

2Dµν (z)

∂

∂νPµν (z) =

e−iπµ cos(πµ)cos2(πν)

[Qµν (z)−Qµ−ν−1(z)

]+ e−iπµ

4π cos(πν)

{sin[π(ν + µ)]Cµ

ν (z)− sin[π(ν − µ)]Cµ−ν−1(z)

}− e−iπµ

2π cos(πν)

{sin[π(ν + µ)]Dµ

ν (z)− sin[π(ν − µ)]Dµ−ν−1(z)

}Dµν (z) = eiπµ(z2 − 1)

µ2

∞∑n=0

Γ( ν+µ+22

+n)Γ( ν+µ+12

+n)n!Γ(ν+ 3

2+n)z2n ψ(ν + 3

2 + n)



What are these Aµν(z), B

µν(z), C

µν(z), D

µν(z)?

most likely independent set of 2 functions

derivatives with respect to parameters of 2F1(a, b; c; z)

∞∑n=0

(a)n(b)nzn

n!(c)nψ(a+ n) = ψ(a)2F1(a, b; c; z) +

∂

∂a2F1(a, b; c; z)

∞∑n=0

(a)n(b)nzn

n!(c)nψ(c+ n) = ψ(c)2F1(a, b; c; z)− ∂

∂c2F1(a, b; c; z)

Brychkov & Geddess (2004) �Di�erentiation of generalized

hypergeometric functions with respect to parameters does not, in

general, lead to generalized hypergeometric functions�

Ancarani & Gasaneo (2008,2009,2010) compute derivatives with

respect to parameters of pFq and show relation to Kampé de Fériet

double hypergeometric series



di�erentiation with respect to parameters of Gausshypergeometric functions in terms of Kampé de Férietmultiple hypergeometric series

∂

∂a2F1(a, b; c; z) =

zb

cF 2:2;1

2:1;0

[a+ 1, b+ 1 : 1, a; 1;2, c+ 1 : a+ 1;−;

z, z

]∂

∂c2F1(a, b; c; z) = −zab

c2F 2:2;1

2:1;0

[a+ 1, b+ 1 : 1, c; 1;2, c+ 1 : c+ 1;−;

z, z

]F p:q;kl:m;n

[(ap) : (bq); (ck);(αl) : (βm); (γn);

x, y

]=

∞∑r,s=0

Qpj=1(aj)r+s

Qqj=1(bj)r

Qkj=1(cj)sQl

j=1(αj)r+sQmj=1(βj)r

Qnj=1(γj)s

xr

r!ys

s!∞∑n=0

(a)n(b)nzn

n!(c)nψ(a+n)=ψ(a)2F1(a,b;c;z)+

zb

cF 2:2;1

2:1;0

[a+ 1, b+ 1 : 1, a; 1;2, c+ 1 : a+ 1;−;

z, z

]∞∑n=0

(a)n(b)nzn

n!(c)nψ(c+n)=ψ(c)2F1(a,b;c;z)+

zab

c2F 2:2;1

2:1;0

24a+ 1, b+ 1 : 1, c; 1;2, c+ 1 : c+ 1;−;

z, z

35Howard Cohl (NIST) parameter di�erentiation April 7, 2011 20 / 31

some particular cases for Legendre functions

Szmytkowski's formulas (see Szmytkowski (2009))

[∂

∂νPmν (z)

]ν=p

= Pmp (z) logz + 1

2

+ [2ψ(2p+ 1)− ψ(p+ 1)− ψ(p−m+ 1)]Pmp (z)

+ (−1)p+mp−m−1∑k=0

(−1)k2k + 2m+ 1

(p−m− k)(p+m+ k + 1)

×[1 +

k!(p+m)!(k + 2m)!(p−m)!

]Pmk+m(z)

+ (−1)p(p+m)!(p−m)!

m−1∑k=0

(−1)k2k + 1

(p− k)(p+ k + 1)P−mk (z)

where p,m ∈ {0, 1, 2, . . .} and 0 ≤ m ≤ p



Szmytkowski's formulas

Some special cases include for m = 0[∂

∂νPν(z)

]ν=p

= Pp(z) logz + 1

2+ 2 [ψ(2p+ 1)− ψ(p+ 1)]Pp(z)

+ 2(−1)pp−1∑k=0

(−1)k2k + 1

(p− k)(p+ k + 1)Pk(z)

and for m = p[∂

∂νP pν (z)

]ν=p

= P pp (z) logz + 1

2+ [2ψ(2p+ 1)− ψ(p+ 1) + γ]P pp (z)

+ (−1)p(2p)!p−1∑k=0

(−1)k2k + 1

(p− k)(p+ k + 1)P−pk (z)

where γ ≈ 0.57721566 is Euler's constant



Application of Szmytkowski's formula

Recently I proved:

‖x− x′‖ν =eiπ(ν+d−1)/2Γ

(d−2

2

)√πΓ(−ν

2

) (r2> − r2

<

)(ν+d−1)/2

(rr′)(d−1)/2

×∞∑λ=0

(λ+

d

2− 1)Q

(1−ν−d)/2λ+(d−3)/2

(r2 + r′2

2rr′

)Cd/2−1λ (cos γ)

for x,x′ ∈ Rd with r = ‖x‖, r′ = ‖x′‖, and r≶ = minmax{r, r

′}, then using

limν→0

∂

∂ν‖x− x′‖ν = ‖x− x′‖2p log ‖x− x′‖

and Whipple's transformation of Legendre functions to obtain a

Gegenbauer expansion for singular logarithmic kernels associated with

fundamental solutions of the even-dimensional polyharmonic equation in

Euclidean spaceHoward Cohl (NIST) parameter di�erentiation April 7, 2011 23 / 31


Cohl's formulas (see Cohl (2010,2011))

In the sequel m,n ∈ {0, 1, 2, . . .}

Γ(ν ∓m+ 12)

Γ(ν −m+ 12)

[∂

∂µPµν−1/2(z)

]µ=±m

= Qmν−1/2(z) + ψ

(ν ∓m+

12

)Pmν−1/2(z)

±m!m−1∑k=0

(−1)k−m(z2 − 1

)(k−m)/2

2k−m+1k!(m− k)P kν+k−m−1/2(z)

Γ(ν ∓m+ 12)

Γ(ν −m+ 12)

[∂

∂µQµν−1/2(z)

]µ=±m

=[iπ + ψ

(ν ∓m+ 1

2

)]Qmν−1/2(z)

±m!m−1∑k=0

(−1)k−m(z2−1)(k−m)/2

k!(m−k)2k−m+1 Qkν+k−m−1/2(z)



Cohl's formulas (see Cohl (2010,2011))

In the sequel m,n ∈ {0, 1, 2, . . .}

±[∂

∂νPµν−1/2(z)

]ν=±n

=[ψ

(µ+ n+

12

)− ψ

(µ− n+

12

)]Pµn−1/2(z)

+n−1∑k=0

n! Γ (µ− n+ 1/2)(z2 − 1

)(n−k)/2

Γ(µ+ n− 2k + 1

2

)k!(n− k)2k−n+1

Pµ+n−kk−1/2 (z)

[∂

∂νQµν−1/2(z)

]ν=±n

= −√π

2eiπµ Γ

(µ− n+

12

)(z2 − 1

)−1/4Qnµ−1/2

(z√

z2 − 1

)

±n!n−1∑k=0

(z2 − 1

)(n−k)/2

2k−n+1k!(n− k)Qµ+k−nk−1/2 (z)


references

Bessel references

A. Apelblat and N. Kravitsky.

Integral representations of derivatives and integrals with respect to the order of the Besselfunctions Jν(t), Iν(t), the Anger function Jν(t) and the integral Bessel function Jiν(t).IMA Journal of Applied Mathematics, 34(2):187�210, 1985.

Yu. A. Brychkov and K. O. Geddes.

On the derivatives of the Bessel and Struve functions with respect to the order.Integral Transforms and Special Functions, 16(3):187�198, 2005.

G. Petiau.

La théorie des fonctions de Bessel exposée en vue de ses applications à la physique mathématique.Centre National de la Recherche Scienti�que, Paris, 1955.

R Müller.

Uber die partielle abietung der besselschen funktionen nach ihrem parameter.Zeitscrift für Angewandte Mathematik und Mechanik, 20(1):61�62, 1940.

G. N. Watson.

A treatise on the theory of Bessel functions.Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1944.


references

Legendre references

Yu. A. Brychkov.

On the derivatives of the Legendre functions Pµν (z) and Qµν (z) with respect to µ and ν.Integral Transforms and Special Functions, 21(3):175�181, 2010.

H. S. Cohl.

Derivatives with respect to the degree and order of associated Legendre functions for|z| > 1 using modi�ed Bessel functions.Integral Transforms and Special Functions, 21(8):581�588, 2010.

H. S. Cohl.

Fourier and Gegenbauer expansions for fundamental solutions of the Laplacian and powers

in Rd and Hd.PhD thesis, The University of Auckland, 2010. xiv+190 pages.

H. S. Cohl.

On parameter di�erentiation for integral representations of associated legendre functions.Submitted., 2011.

L. Robin.

Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome II.Gauthier-Villars, Paris, 1958.


references

Legendre references (cont.)

R. Szmytkowski.

Addendum to: �On the derivative of the Legendre function of the �rst kind with respect to its degree�.Journal of Physics A: Mathematical and Theoretical, 40(49):14887�14891, 2007.

R. Szmytkowski.

A note on parameter derivatives of classical orthogonal polynomials.ArXiv e-prints, 0901.2639, 2009.

R. Szmytkowski.

On the derivative of the associated Legendre function of the �rst kind of integer degreewith respect to its order (with applications to the construction of the associated Legendrefunction of the second kind of integer degree and order).Journal of Mathematical Chemistry, 46(1):231�260, June 2009.

R. Szmytkowski.

On the derivative of the associated Legendre function of the �rst kind of integer orderwith respect to its degree (with applications to the construction of the associated Legendrefunction of the second kind of integer degree and order).ArXiv e-prints, 0907.3217, 2009.

R. Szmytkowski.

Green's function for the wavized Maxwell �sh-eye problem.Journal of Physics A: Mathematical and Theoretical, 40(6):065203, 2011.


references

Hypergeometric references

L. U. Ancarani and G. Gasaneo.

Derivatives of any order of the con�uent hypergeometric function 1F1(a, b, z) with respect to theparameter a or b.Journal of Mathematical Physics, 49(6):063508, 16, 2008.


Derivatives of any order of the Gaussian hypergeometric function 2F1(a, b, c; z) with respect to theparameters a, b and c.Journal of Physics. A. Mathematical and Theoretical, 42(39):395208, 10, 2009.


Derivatives of any order of the hypergeometric function pFq(a1, . . . , ap; b1, . . . , bq ; z) with respectto the parameters ai and bi.Journal of Physics. A. Mathematical and Theoretical, 43(8):085210, 11, 2010.

Yu. A. Brychkov and K. O. Geddes.

Di�erentiation of hypergeometric functions with respect to parameters.Abstract and applied analysis, 15�28, World Sci. Publ., River Edge, NJ, 2004.

H. M. Srivastava and Per W. Karlsson.

Multiple Gaussian hypergeometric series.Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester, 1985.


references

Archival references

M. Abramowitz and I. A. Stegun.

Handbook of mathematical functions with formulas, graphs, and mathematical tables,volume 55 of National Bureau of Standards Applied Mathematics Series.U.S. Government Printing O�ce, Washington, D.C., 1972.

Yu. A. Brychkov.

Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas.Chapman & Hall/CRC Press, Boca Raton-London-New York, 2008.

I. S. Gradshteyn and I. M. Ryzhik.

Table of integrals, series, and products.Elsevier/Academic Press, Amsterdam, seventh edition, 2007.Translated from the Russian, Translation edited and with a preface byAlan Je�rey and Daniel Zwillinger, With one CD-ROM (Windows, Macintosh and UNIX).

W. Magnus, F. Oberhettinger, and R. P. Soni.

Formulas and theorems for the special functions of mathematical physics.Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52.Springer-Verlag New York, Inc., New York, 1966.

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.

NIST handbook of mathematical functions.U.S. Department of Commerce, National Institute of Standards and Technology, Washington, D.C., 2010.With 1 CD-ROM (Windows, Macintosh and UNIX).


summary

Future work in this classical area of Legendre functions

organize the known results, which are scattered, for parameter

derivatives in particular cases

�nd connections and pursue properties of Aµν ,B

µν ,C

µν ,D

µν functions

study multiple derivatives with respect to parameters

investigate anti-derivatives with respect to parameters


parameter differentiation for bessel and associated legendre functions › ~dlozier › sf21 ›...

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