Special topics in the quantum theory of angular momentum

Pram~r~a, Vol. 24, Nos 1 & 2, January & February 1985, pp. 15-26. © Printed in India.

S p e c i a l topics in the q u a n t u m theory o f a n g u l a r m o m e n t u m K SRINIVASA RAO

The Institute of Mathematical Sciences, Madras 600113, India

Abstract. Two special topics in the quantum theory of angular momentum are discussed in this article. They are: (i) the relationship between the coupling and recoupling coefficients (for two and three angular momenta, respectively) and sets of generalized hyper-geometric functions of unit argument; and (ii) the 'non-trivial' or polynomial zeros ofangular momentum coefficients and their classification.

Keywords. Quantum theory; angular momentum; coupling coefficients; recoupling coef- ficients; hypergeometric functions.

PACS No. 02.90; 03.65; 21-90

1. Introduction

The new developments in the quantum theory of angular m o m e n t u m , during the past 25 years following the dramatic discovery o f new symmetry properties o f the angular m o m e n t u m coupling and recoupling coefficients (for two and three angular m o m e n t a , respectively) by Regge (1958, 1959) led to the subject itself becoming a part o f the

Encyclopedia of Mathematics and its Applications.

Volumes 8 and 9 o f this series, by Biedenharn and Louck (1981) are entitled, "Angular m o m e n t u m in q u a n t u m physics"

and " T h e Racah-Wigner algebra in q u a n t u m theory", respectively. In C h a p t e r 5, o f Vol. 9, twelve topics are developed and these establish the diverse relationships between concepts in angular m o m e n t u m theory and other areas o f mathematics. T h e new developments, in the words o f Smorodinskii and Shelepin (1972)are " m o r e and m o r e intertwined with various sections o f algebra, multidimensional geometry, topology, projective geometry, analytic function theory, the theory o f special functions, differen- tial equations, combinatorial analysis and the calculus o f finite differences. O n e could say that the theory o f Clebsch-Gordan coefficients takes on the character o f a new kind o f calculus, going far beyond the scope o f the classical theory".

In this article, we devote our attention to only two special topics o f recent origin, where we have made some significant contributions. They are: (i)the relationship between the coupling (Clebsch-Gordan) and recoupling (Raeah) coefficients and sets o f generalized hypergeometric functions o f unit argument, and (ii) the 'non-trivial' or polynomial zeros o f the Clebsch-Gordan (3-j) and Racah (6-j) coefficients and their classification.

15

2. T h e relationship between the C i e b s c h - G o r d a n and R a c a h c o e f f i c i e n t s and genera- iized h y p e r g e o m e t r i c functions

2.1 The

Clebsch-Gordan coefficient

Here we show that the series which is given in literature (see for instance, Biedenharn and Louck 1981) for the Clebsch-Gordan or 3-j coefficient belongs to a set o f six representations and that there exist, correspondingly, a set o f six 3F2(1)s, which is necessary and suffÉcient to account for the 72 symmetries o f the 3-j coefficient.

The 3-j coefficient is defined by:

J,

J2 J3

~ = 6(ml + m2 + m3)( _ 1 ) J , - - m 3 _~j2 A ( j I J2 J 3 ) m l m 2 m3 J

3

× I'l [(ji+mi)! (ji--ml)!] 1/2 i=1

f 2 3 ]-'

x ~ ( - 1 ) ' t! n ( t - ~ ) ! H ( B , - O ! ,

O)

k = l I=1

where

max(aq, ~t2) ~< t ~< min(fll, f12, f13),

fix = Jx -- m l , f12 = J2 + m2, fla = Jl +J2 --J3,

~1 = Jl --J3 + m2 = Jl -- ml -- (J3 + m3), or2 -- J2 --J3 -- ma = J2 -t- m 2 -- (J3 -- m3),

andA(xyz)

= [ ( x + y - z ) !

( x - y + z ) ! ( - x + y + z ) ! / ( x + y + z +

1)!] t/2. Equation (1)is invariant to permutations o f the three angular momenta (column permutations) and to space reflections (m i ---, - m l ) , thereby exhibiting a 12-element symmetry group.

Regge (1958) arranged the nine integer parameters referred to by Racah (1942), into a 3 x 3 square symbol and represented the 3-j coefficient as:

mlm2m3 /

J x - m l J 2 - m 2 J 3 - m 3 ---IlRik II. (2) Jx + ml J2 + ma Js + ms

The fact that all sums o f columns and rows are equal to J = Jl -l-J2 +J3 leads to the following nine relations amongst the elements of

Rip + Rmp :

R.q + R.,,

(3)

for cyclic permutations o f both

(Iron)

= (123) and

(pqr)

= (123). Regge (1958) dis- covered that there exists a 72-element symmetry group, comprising the aforementioned 12-element symmetry group, for the 3-j coefficient, due to the invariance o f IIRit If to its column and row permutations and a reflection about its diagonal.

Racah (1942) pointed out that substitution for the summation index t, the argument o f one o f the five factorials in (1) only leads to some symmetry properties o f the angular m o m e n t u m coupling coefficient. It is straightforward to show that the five substitution procedures lead to five series representations. We show here that these together with (1) constitute a set of six series representations for the 3-j coefficient, which is necessary and sufficient to account for its 72 symmetries. Since the 72-element symmetry group is evident when the 3-j coefficient is represented by II R~k II, it is worthwhile to note that the

Quantum theory of angular momentum 17

set o f six series representations can be written in the following compact notation:

"3

[[Rik[[

=t~(m 1 + m 2 + m 3 ) rI

[Rik!/(J + l)] 1/2

i , k = l

x ( -- 1) °~pq') ~ Is ! (R2p - s)! (R3q - s) ! (R 1, -- s)!

$

x (s + R 3 , - R2p)!

( s + R 2 , -

Raq)!] -1 (4) for all permutations o f

(pqr) =

(123), with

Rap--R2q for even permutations o f (123)

O(pqr) = R a p - R 2 q + J

for odd permutations o f (123).

Let the six series representations be denoted by the R o m a n numerals (I) to (VI), corresponding to the six permutations o f

(pqr)

in (4): (123), (231 ), (312), (132), (321) and (213), respectively. This set of six series representations can also be obtained by permuting the indices (123) in (1), and remembering that the series acquires an additional phase factor o f ( - 1) ~, for odd permutations. In establishing the one-to-one correspondence between the series obtained by the substitution procedures and that obtained by permuting the indices in (I), use is made o f the fact that 4ji is an even integer so that ( - 1) 4j, is always positive. The following observations can now be made, with the help o f (3) where necessary:

(a) the column permutations o f

I[Rik

[[ are in one-to-one correspondence with the six series representations, thereby spanning the whole set given by (4);

(b) the permutation o f the second and third rows o f [[Rik [[ which corresponds to space reflection, exchanges (I) ~ (VI), (II) ~-~ (V) and (III) ~-~ (IV);

(c) the permutation o f the first and second rows o f

[[Rik

[[ exchanges ( I ) ~ (V), (II) ~-~ (IV) and (III) ~ (VI);

(d) the permutation o f the first and third rows o f [[Rik[[ exchanges (I)~-~ (IV), (II) ~ (VI) and (III) ~-, (V);

(e) the cyclic permutation o f rows, (123) ~ (231), o f

[[Rik

[[ permutes anti-cyclically the first three and the last three o f the six series representations, amongst themselves, as: (I)

(III) - , (II) ~ (I) and (IV) -~ (VI) ~ (V) ~ (IV);

(f) the cyclic permutation o f rows, (123) ~ (312), o f

[[Rig

[[ permutes cyclically the first three and the last three o f the six series representations, amongst themselves, as: (I) -~ (II) ~ (III) ~ (I) and (IV) --* (V) --* (VI) --, (IV);

(g) the reflection o f

[[Rik [[

about its diagonal interchanges (I) ~ (III) leaving the other four series representations invariant;

(h) the reflection o f

HRik [[

about its skew-diagonal interchanges (IV) ~ (V) leaving the others invariant;

(i) each series representation is left invariant to the combined operation o f an odd column permutation and the space reflection; as well as to either the Regge'symmetries (Appendix 1) as such or Regge symmetries on which are superposed an even column permutation or an odd column permutation and the space reflection;

P 2

(j) each series representation exhibits 12 of the 72 distinctly different symmetries of the 3-j coefficient and this 12-element symmetry group is isomorphic to the product of permutation groups of 3 objects (R2p,

R3q,

R1, ) and 2 objects ( R 3 , - R 2 p , R 2 , - R 3 q ).

From the observations (a) to (j) it follows that the set of six series representations is necessary and sufficient to account for the 72 known symmetries of the 3-j coefficient.

Conventional ways (refer, for instance, deShalit and Talmi 1963) of establishing the symmetry relations are tedious and some (Rose 1955) necessarily resort to the substitution procedures. Our observation that

IIRik II

is in fact a set of six series representations enables one to establish the symmetries easily. Notice that the first three series corresponding to even column permutations are separated from the last t h r e e series which correspond to odd column permutations, in the sense that the latter contain the additional phase factor, ( - 1) J.

The series representations can be rearranged into generalized hypergeometric functions (Slater 1966) of unit argument by using:

n! = F(n + 1) (5)

for the factorials and whenever the summation index in the argument o f the F function is negative resorting to:

F (n) F (1 - n) = n cosec nn (6)

to replace it by a F-function with an argument containing a positive index o f summation. This procedure of rearrangement yields in the case of the 3-j coefficient (Srinivasa Rao 1978; Venkatesh 1978) the following set of six

3Fz(1)s:

IlRi~ll -- 6(ma + m 2 +m3) Rik!/(d +

1)

i, 1

× ( - 1)",,')IF(1 - - a , 1 - B , 1 - C , D, E)] -~

where

x 3F2 (ABC; DE;

1), (7)

A = - - R 2 p , B = - R 3 q , C = - R I , , D = 1 + R 3 , - R2~,,

E = I + R 2 , - - R 3 q

and

F(x,y . . . .

) = F ( x ) F ( y ) . . . .

for all permutations of

(pqr)

--- (123). This set of six 3F2 (1)s given here are all of the van der Waerden's form (Smorodinskii and Shelepin 1972)----the 3F2(1) given by (5.21) of this reference corresponds to

(pqr) =

(123).

2.2

The Racah coefficient

Here we show that it is possible to obtain two equivalent sets of series representations for the Racah (6-j) coefficient, which can be rearranged into two equivalent sets of 4F3(1)s. These sets are shown to be related to each other through the property of reversal of series of the generalized Saalschutzian hypergeometric function.

The conventional expression for the Racah coefficient (Biedenharn and Louck 1981) is given by:

{ a b f } dc = ( -

1)a+b+c+ ,

W (abcd;ef)

Quantum theory o f anoular momentum 19

= N ~ ( -- I)P(P + 1)! (P--0q)! ~ - P ) ! , (8)

p i j = l

where the range o f P is Pmi. ~< P ~< Pm~, with Pmi. = max(tel, ~2, ~3, ~4),

Pmax = min (fit, f12, fla),

cx t = a + b + e , c~ 2 = c + d + e , ~ 3 = a + c + f , ~ 4 = b + d + f , f l i = a + b + c + d , f l 2 = a + d + e + f , f l a = b + c + e + f , N = A (abe) A (cde) A (acf) A (bdf),

and A(xyz) defined as before (below equation (1)).

The 6-j coefficient is invariant under:

(i) the 3! ( = 6) column permutations and

(ii) the four interchanges o f any two elements in the first row o f th¢6-j symbol with the corresponding elements in the second row (this will be referred to as 'row' permutation in the text). These 24 symmetries constitute the classical tetrahedral symmetry group o f the 6-j coefficient.

In passing it is to be noted that this series form (8) can be found for the first time in Regge's (1958) article. Racah (1942) dealt with only the series expansion obtained by substituting s = fl] - P in (8), which corresponds to one o f a set o f three represen- tations, to be discussed later.

Once the series is in the form (8), it is easy to see that the 6-j coefficient exhibits the 144-element symmetry group, due to its invariance under the permutation o f the four

• 's and the three fl's. It was Regge (1958) who dramatically discovered six more symmetries (Appendix 2) and established that the Racah (6-j) coefficient exhibits 144 symmetries and n o t only the 24 tetrahedral symmetries.

By setting in (8), s = f l k - P , k = l, 2, 3, in succession, we get the set I o f three series expansions:

{abe =

d c f J N ( - l ) ~ . ~ ( - l ) ' ( ~ - s + l ) ! ,

x (ilk -- Oq -- S)! (S + flk -- flS)! • (9)

i = 1 j = l

Notice that a series belonging to this set I exhibits only 48 o f the 144 symmetries, due to the permutation o f all the four ~'s but only two o f three fl's, since flk is now in the numerator in (9).

Instead, if we set in (8), s = P - ~z, 1 -- 1, 2, 3, 4, in succession, we obtain (Srinivasa Rao and Venkatesh 1977) the following set II o f four series representations:

f:

^{c f} = N ( - I ) ' , ~ ( - 1 ) * ( a l + s + 1)!

~t

× H l-I (1o)

/ = 1 j = l

where we notice that a series belonging to set II (10) exhibits only 36 o f the 144

symmetries, arising due to the permutation of all the three fl's, but only three of the four

~t's since 0q is now in the numerator in (10).

Thus, the 144 symmetries of the Racah (6-j) coefficient are exhibited by the single series expansion (8), or by the set I of three series representations given by (9), or, equivalently, by the set II of four series representations given by (10).

When (9) is rewritten in terms of generalized hypergeometric functions of unit argument, we get (Srinivasa Rao et a11975), the set I of three generalized Saalschutzian hypergeometric functions of unit argument:

{

a b e ~ =

d c f J ( - 1 ) r + I N F ( 1 - E ) [ F ( 1 - A , 1 - B , 1 - C , 1 - D , F , G ) ] -1

x 4F3(ABCD; EFG; I), (11)

where

A = e - - a - b , B = e - c - d , C = f - a - c , D = f - b - d , E = - a - b - c - d - l , F = e + f - a - d + l , G = e + f - b - c + l

and F ( x y z . . . . ) = F(x) F(y) F(z) . . . . Superposing the column permutations of

{ a b e ~

4~ys on the 4Fa(1 ) in (11), yields us the set I of three 4Fa(1)s. We note that the superposition of a 'row' permutation of { ~ ~ ~ } on the 4Fa (1) in (11) results only in a permutation o f the numerator and denominator parameters amongst themselves in a given 4Fa(1 ) belong to this set I.

When (10) is rearranged into a set of hypergeometric functions, we obtain (Srinivasa Rao and Venkatesh 1977) the following set II of four 4Fa(1)s for the 6-j coefficient:

~ a b e ~ = _ _ . . . . .

~ d c f J ( B' 1 - c , 1 - - D , E , F , G')] -1

x ,,Fa(A'B'C'D'; E'F'G'; 1), (12)

where

A' = a + c + f + 2 , B' = c - d - e , C' = a - b - e , D' = f - b - d , E' = a + c - b - d + 1, F' = a + f - d - e + 1, G' = c + f - b - e + 1.

The set of four 4F3 (1)s is spanned by superposing the interchange of'row' permutations

a b e

of { a ~ s } on the given 4F3 (1) in (12). Superposing the column permutations of the 6-j coefficient on a given 4F3 (1) belonging to this set results only in a permutation of the numerator and denominator parameters amongst themselves.

Since more than one numerator parameter in the set I and set II of 4F3 (1)s is a negative integer, we can generalize the property of 'reversal of the series' given by Bailey (19'35) for the case of a 3F2(1), to the case of a Saalschutzian 4F3(1) to obtain the identity:

4Fa(ABCD; EFG; 1) = ( - 1 ) D F ( 1 - A , l - B , 1 - C , F, G, D - E + I ) x [ F ( D - A + I , D - B + I , D - C + I , F - D , G - D , I - E ) ] -1

x 4 F a ( D - E + I , D - F + I , D - G + I , D ; D - A + I , D - B + I , D - C + I ; 1 ) , (13) where D is the negative parameter which determines the number of terms in the series.

Using (13) in (11), after some simple algebraic manipulations, we get:

{ a b e ~ = , ,

d c f J

E', 1 - C , 1 - B , 1 - D ' ) ] - '

x aF3(A'C'B'O'; F'G'E'; 1), (12')

Quantum theory o f anoular momentum 21 which is nothing but the 4F3(1) belonging to set II given by (12) except for a trivial permutation amongst the numerator and denominator parameters o f the generalized hypergeometric function.

Thus, we find that the set I and set II of 4F3(l)s for the 6-j coefficient are not independent but are related to one another by the property o f reversal of series. It is intriguing to note that due to reversal of series property the set I of three 4F 3 (1)s yields the set II of four 4F3 (1)s and vice versa.

It has been possible for us (Srinivasa Rao and Venkatesh 1978; Srinivasa Rao 1981) to show the advantages of using the sets of pF~ (1)s in the numerical computation o f the 3-j and the 6-j coefficients. The pFq(l)s can be computed numerically using Horner's rule (Lee 1966) for polynomial evaluation as:

~ F ~ ( ~ . . . ~ ; / h # 2 - - . / ~ ; z)

...], ^,,4,

where

P q

x i = I-I (~j+i) and Y i = ( i + l } I-I (fl~+i)"

j = l k = l

The use of (14) places our approach of numerical computation on the same footing as that o f Wills (1971) and in the nested form the number of multiplications is minimum.

In fact, the expression of Bretz (1976) is identical to the set II of 4F3(1)s given by (12).

Thus, the set of ~7~(1)s which have been shown to be necessary and sufficient to account for the symmetries o f the 3-j and the 6-j coefficient have been found to be not only useful but also time-saving when compared to the best available computer (Fortran) programs of Wills (1971) and Bretz (1976) by about 5 to 15 %.

D'Adda et al (1974) and Raynal (1978, 1979) make use o f Whipple's (1925, 1936) work on the symmetries of 3F2 functions, well-poised 7F6 functions and Saalschutzian 4F3 functions, all with unit argument, to study the 3-j and the 6-j symbols generalized to any arguments. Raynal (1978) generalized one o f the formulas for the 3-j symbol of SU (2)to obtain a generalized 3F2 (1) with complex parameters. Biedenharn and Louck (1981), discuss this aspect of the Racah-Wigner algebra--viz the relationship of the 3-j and the 6-jcoefficients to generalized hypergeometric functions--as a special topic in Vol. 9 of the Encyclopedia in Mathematics and its Applications and summarize the results of our work presented here in a different notation.

3. 'Non-trivial' or p o l y n o m i a l zeros o f angular m o m e n t u m coefficients and their classification

Here we show that the polynomial or 'non-trivial' zeros of the Clebsch-Gordan (3-j) and the Racah (6-j) coefficients can be classified according to their degree. The majority of the zeros tabulated to-date, hitherto considered as non-trivial, are in fact polynomial zeros of degree one, arising due to the existence of binomial expansions for these coefficients.

3.1 Clebsch-Gordan coefficient

Trivial zeros of the 3-j coefficient are those which arise due to either the triangular inequality not being satisfied byj~,j2 and J3; or due to d ( = Jl +J2 +J3) being odd when

n h = m 2 = m 3 = 0 (which can be easily verified as the consequence o f the symmetry property o f the 3-j coefficient under spatial reflection: ms ~ -ms). The 'non-trivial' zeros are identified as the zeros of the polynomial part of the 3-j coefficient, defined in (4). They are also called as "structural" zeros.

Sato and Kaguei (1972), used the concept of generalized powers to obtain a formal binomial expansion for the 3-j coefficient. Following the procedure given by Sato (1955) and using the symbolic notation for generalized powers:

ptx~ = p ! / ( p _ x)!, (15)

we can rearrange the set of six series representations given by (4) into the form:

3

IIR,kil

= 6(ml + mz + m3) I-I [R,k [ / ( J + 1 ) ] 1/2 ( -- 1) °tpa')

i,k

x I F ( n + 1, Cu+ 1, Cv+ 1, B , p + n + 1, B r ~ + n + 1)] - t

x { (Brp + n) (B,~ + n) - Cu C v } "), (16) where n = min(Rzp, 3q, R~,), C~, Cv represent the Rik's in the triple (Rzp , R3~, R~,) other than the minimum, Brp = R 3 , - Rzp and B,q = R z , - Raq. The expression (16) is a generalization of the formal binomial expansion obtained by Sato and Kaguei (1972), for their result can be obtained by putting (pqr) = (231) in (16), while (16) itself holds for all the six permutations of (pqr) = (123).

Obviously, since the generalized power (15) is exact for n = 1, (since Pt~) = P) the binomial form for the 3-j coefficient explicitly reveals further structure zeros of this coefficient. These can be shown (Srinivasa Rao and Rajeswari 1984a) to be polynomial zeros o f degree one. We can simply locate all the polynomial zeros of degree one of the 3-j coefficient, independent of the numerical values of the arguments of the 3-j coefficient, with the help of the simple factor:

(1 - ' L . , 'L.0, (17)

where x and y are given by:

x = R,~ Rkp and y = Rmp

Rkr,

with (lmk) and (pqr) corresponding to specific permutations of (123) when n = Rl~.

Polynomial zeros of the 3-j coefficient were listed, for J ( = Jl +J2 +Ja) ~ 27 by Varshalovich et al (1975). Bowick (1976) used the Regge symmetries for the 3-j and the

64

coefficients to obtain listings of only the Regge-inequivalent polynomial zeros. The number of terms in the polynomial part of the 3-j coefficient is governed by: n

= min(R2p, R3~, Rt,) and we use this simple prescription to classify the polynomial zeros tabulated by Bowick (1976) and by Biedenharn and Louck (1981) into those which correspond to degree 1, 2 and 4. The classified tables can be found in Srinivasa Rao and Rajeswari (1984b). From the tables, we conclude that for J ~< 27, of the 36 polynomial zeros, 21 are polynomial zeros of degree 1, revealed by the exact binomial expansion for this coefficient, represented by the multiplicative factor (17).

3.2 The Racah coe~icient

Trivial zeros of the Racah coefficient are those which arise due to a violation of one of the four triangular inequalities to be satisfied by the six angular momentum arguments.

Quantum theory o f anoular momentum 23 The existence o f a class o f zeros o f the 6-j coefficient, not due to the aforesaid reason has been called as polynomial, 'non-trivial' or structure zeros by Koozekanani a n d Biedenharn (1974).

Sato (1955) expressed the Racah coefficient symbolically as a binomial expansion.

Following his notation but making a different set o f substitutions, corresponding to those required for getting the set I o f series expansions given by (9), given below:

s =/~o - P , 0 ~< s ~< n, n = / / 0 - % ,

Ai = flo --~, (i = p,q,r), Bj = f l i - f l o (J = u, v), (18) where the indices p, q, r and u, v are used for those values o f the ~'s and fl's other t h a n Pml~ = max (~1, %, ~3, ~4) = % and Pmax = min(fll, f12, f13) = flo, we obtain (Srinivasa Rao a n d Venkatesh 1977) the binomial expansion:

{

a b e ~ = N t - 1 ) # o [ F ( n + I , A p + l , A , + 1, A , + 1, B , , + n + l , B , , + n + l ) ] -1 de f }

× { (B u + n) (B v + n) - Af4qA,(~o + 1) (- 1)} (.) (19)

where we used the notation:

po,) = P ! / ( P - x ) ! and P~-~)= 1/P (')

and regarded the form P~) to represent the generaliTed powers o f P. It follows that (19) is an exact binomial expansion for the power n = 1. The structure zeros which arise f r o m the polynomial part o f the 6-j coefficient are indeed polynomial zeros o f degree one. All these polynomial zeros o f degree one are accounted for by the simple factor:

(1 - ~ix,r 6.. ~ ), (20)

where

X = ( B . + n ) ( B v + n ) ( f l o + 1),

=

(fl,,--%)(fl,,-%)(flo+

I), and

Y = Ap Aq A,,

= (//o - % ) (/~o - ~ ) ( ~ o - oc,). (21)

~JlJ2J3~

Koozekanani and Biedenharn (1974) calculated the 6-j coefficient [1112 l~ J for arguments j~, l~ ~< 18.5 for i = 1, 2, 3 and found its polynomial zeros. Using the symmetries o f the 6-j coefficient, they ordered the arguments Jl, J2, J3, I1, 12, I3, in a speedometric fashion with Jl the slowest varying and 13 the most rapidly changing variable. The vanishing values o f the 6-j coefficient were calculated by a computer program which resorted to the use o f numbers decomposed into powers o f prime factors. Bowick (1976) used the Regge symmetries (Appendix 2) o f the 6-j coefficient to obtain a reduced listing o f the Regge-inequivalent polynomial zeros for arguments

~< 18-5.

O n the other hand, we have used a simple program which cheeks for the multiplicative factor (20) being equal to zero, to find all the polynomial zeros o f degree one, forj~, l~ ~< 18.5 for i = 1, 2, 3. Having separated the majority o f the zeros, we sorted out the remaining polynomial zeros tabulated by Koozekanani and Biedenharn (1974), according to their degree given by n =.fl0 - %- These tables can be found in Srinivasa

Rao and Rajeswari (1984b). From our classified tables we conclude that 1174 out of 1420 polynomial zeros tabulated by Koozekanani and Biedenharn (1974) are polynomial zeros of degree one, revealed by (20) which is a consequence of the exact binomial expansion (19).

We have also shown (Srinivasa Rao 1985) that of the 12 generic or Regge- inequivalent zeros which have been explained to be due to either triangle rule violation for quasi-spin (Koozekanani and l]iedenharn 1974), or vanishing of fractional parentage coefficients in the atomic g-shell (Judd 1970), or realizations of exceptional Lie algebras G 2, F 4 and E 6 (Van der Jeugt et a11983; Van den Berghe et a11984}--eleven are trivial polynomial zeros of degree one. We also conjectured (Srinivasa Rao 1984) that, in principle, one can find closed form formulas for the polynomial zeros of the 3-j and the 6-j coefficients, provided we look upon these coefficients as generalized hypergeometric functions of unit argument, which are analytic, and extend the method o f Siewert and Burniston (1972) to determine zeros of analytic functions to the case of analytic pFq(l)s.

Acknowledgement

The author would like to record his indebtedness to Dr R Ramanna for his constant encouragement and interest and he deems it a pleasure and a privilege to contribute this article to the special issue of this journal to mark his 60th birthday.

Appendix 1

Regge (1958) symmetries for the 3-j coefficient, written down explicitly (Srinivasa Rao 1978) are:

( Jl J2 J3 )=(Jl

ml

m2 m3 J2 --J3 ½(J2Wj3+ml) ½(J2+J3--ml) )

½(J3--j2+ml)+m2 ½(J3--j2+ml)+m3

=(~(Jl+j3+m2) J2 ½(Jl+J3--m2))

(J3 --Jl +m2)+ ml Jl --J3 ½ (J3 --Jl + m2) -I'- ~r/3

=(~(Jl+J2--m3) ½(Jl+j2+m3) J3 )

(Jx . j 2 + m 3 ) + m l

½(Jl --j2+m3)+m2

J2 --Jl '

= :½ (Jl +J2 --m3) ½ (J2 +J3 --ml) ½ (J~ +J3 --m2)

\j3-½(j~+j2+m3) Jl--½(J2+j3+mz) j2-½(jt+j3+m2) )

1 (Jl +J2 + m3) ~ (J2 +J3 + ml )

=

(Jl +J2 --m3)--J3 ½ (J2 +J3 --ml)--Jl

½(Jl +J3 +rn2)

½ (Jl +Ja -m2) -J2

,]"

Only the first and fourth of these symmetries are essentially new, the others can be obtained from these and the column permutations and space reflection symmetries. For an interpretation of the symmetry of the Clebsch-Gordan coefficients discovered by Regge, refer Bincer (1970)

Quantum theory o f anoular momentum 25 Appendix 2

The Regge (1959) symmetries for the Racah coefficient are:

{abf} {: ½ ( b + c + e - f ) ½ ( b - c + e + f ) } dc = 12(b+c-e+f) ½(-b+c+e+f)

{~ ^(a-d+e+f) b ½(a+d+e-f)~

(-a+d+e+f) c ½(a+d-e+f)) {~(a+b+c-d) ½(a+b-c+d) e}

=

(-a+b+c+d) ~(a-b+c+d) f

{~ (b+c+e-f) ~(a-d+e+f) ½(a+b-c+d)~

= (b+c-e+f) ½(-a+d+e+f) ~(a-b+c+d)J {~(b-c+e+f) ½(a+d+e-f) ~(a+b+c-d) }

=

(-b+c+e+f) ½(a+d-e+f) ½(-a+b+c+d) "

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