Normal-ordered equivalent of the Weyl ordering of $\hat{q}^j \hat{p}^k$

Normal-or dered equiv alent of the W e y l ordering of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 Hendry M. Lim 1, 2 , ∗ 1 Resear c h Cent er f or Quantum Physics, National R esearc h and Innov ation Ag ency (BRIN), South T ang erang 15314, Indonesia 2 Department of Physics, F aculty of Mathematics and Natur al Sciences, U niv ersitas Indonesia, Depok 16424, Indonesia The problem of quantizing a bivariate dynamical sys tem can be reduced to ev aluating the order ing of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . Here, we consider the W ey l ordering of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 that is then expressed in term of the annihilation ˆ 𝑎 and creation ˆ 𝑎 † operator . The e xplicit f ormula for the normal-ordered equivalent (all ˆ 𝑎 † ’ s preceeding all ˆ 𝑎 ’s) of the resulting e xpression is then giv en, and some relations are discussed. I. INTR ODUCTION A biv ariate dynamical sys tem is typicall y described in a 2-dimensional phase space by the canonical position 𝑞 and canonical momentum 𝑝 , endo wed with the Poisson brack et { 𝑞 , 𝑝 } = 1. In an attempt to quantize (i.e., build a q uantum mechanical description of ) the sys tem, we replace 𝑞 b y its operator equivalent ˆ 𝑞 , and 𝑝 b y ˆ 𝑝 , defining a Hilber t space endow ed b y the commutator [ ˆ 𝑞 , ˆ 𝑝 ] = 𝑖 ℏ . Doing this, ho w ev er , comes with a problem: quantizing ter ms like 𝑞 𝑝 introduces ordering ambiguity; w e may ha v e ˆ 𝑞 ˆ 𝑝 , or ˆ 𝑝 ˆ 𝑞 , or 𝜋 ˆ 𝑞 ˆ 𝑝 + ( 1 − 𝜋 ) ˆ 𝑝 ˆ 𝑞 , etc. The obstr uction to q uantization culminates in the no-go theorem of Groenew old and V an-Ho v e, which states that there is no quantization map that can globall y preserve the P oisson brac ket structure [ 1 ]. Ne vertheless, quantization of a classical sy stem does result in a quantum sy stem that may reasonabl y mimic the classical sys tem [ 2 – 4 ], making the procedure worth doing in theoretical approaches. In this work, w e consider the combinatorics from choosing the W ey l ordering dur ing the quantization procedure [ 5 ]. This ordering is par ticular ly impor tant since it cor responds to the paradigmatic Wigner representation of quantum sy stems in phase space [ 6 ], forming the so-called Wigner - W e yl cor respondence. W e show that the analy sis of W ey l-ordered quantization f or a bivariate dynamical sys tem can be reduced to ev aluating the W e y l order ing of the operator ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . Our contribution is to e xpress the W ey l order ing in terms of the widely -used annihilation ˆ 𝑎 and creation ˆ 𝑎 † operator , then re wr ite the result into its normal-ordered equivalent, where all ˆ 𝑎 † ’ s precede all ˆ 𝑎 ’ s in each term. II. DISCUSSION A. Quantizing a bivariate dynamical system boils do wn to ordering ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 A general biv ar iate classical sy stem is defined by the eq uations 𝑞 ′ = 𝐴 ( 𝑞 , 𝑝 ) ; (1a) 𝑝 ′ = 𝐵 ( 𝑞 , 𝑝 ) , (1b) where the prime denotes differentiation with respect to time. W e consider the case where 𝐴 and 𝐵 are polynomials in 𝑞 and 𝑝 , 𝐴 ( 𝑞 , 𝑝 , 𝑡 ) = ∞  𝑗 = 0 ∞  𝑘 = 0 A 𝑗 𝑘 𝑞 𝑗 𝑝 𝑘 , (2) and like wise f or 𝐵 . Formally , quantizing this sy stem means finding a quantum sy stem that is expected to generate the dynamics. The quantum expected dynamics cor responding to the classical equation of motion is obtained b y replacing 𝑞 and 𝑝 with the corresponding quantum expectation v alues ⟨ ˆ 𝑞 ⟩ and ⟨ ˆ 𝑝 ⟩ , ⟨ ˆ 𝑞 ⟩ ′ =  ˆ 𝐴  W ; (3a) ⟨ ˆ 𝑝 ⟩ ′ =  ˆ 𝐵  W , (3b) ∗ Current affiliation: Department of Materials Science and Engineering, N ational Univ ersity of Singapore, Singapore 117575, Sing apore; Email: hendry .minfui.lim@u.nus.edu 2 where w e hav e kept the hats inside the bra-kets to emphasize the noncommuting proper ties of the operators in v olv ed. In this w ork, w e specifically consider the W ey l order ing, as indicated by the subscript W . In W ey l-ordered quantization, an e xpression is mapped to the a v erage of all possible operator orderings; f or e xample, 𝑞 𝑝 is mapped to ( ˆ 𝑞 ˆ 𝑝 + ˆ 𝑝 ˆ 𝑞 ) / 2 [ 7 ]. W e hav e  ˆ 𝐴  W = ∞  𝑗 = 0 ∞  𝑘 = 0 A 𝑗 𝑘 𝑁 𝑗 𝑘 *  𝜔  ˆ 𝑞 𝑗 ˆ 𝑝 𝑘  𝜔 + , (4) and lik e wise f or  ˆ 𝐵  W . Here 𝑁 𝑗 𝑘 = ( 𝑗 + 𝑘 ) ! / 𝑗 ! 𝑘 ! is the number of possible orderings 𝜔 of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . W e can treat each term in the series on its own, whence 𝑁 − 1 𝑗 𝑘 Í 𝜔  ˆ 𝑞 𝑗 ˆ 𝑝 𝑘  𝜔 is just the W e yl ordering of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . B. The ladder operators T ypically , it is practical to introduce the annihilation ˆ 𝑎 and creation ˆ 𝑎 † operators b y ˆ 𝑎 = ˆ 𝑞 + 𝑖 ˆ 𝑝 √ 2 ℏ , (5) satisfying  ˆ 𝑎 , ˆ 𝑎 †  = 1. These operators are useful as man y quantum mechanical models can be descr ibed as modifications to the quantum simple har monic oscillator [ 2 – 4 ], for which ˆ 𝑎 annihilates one quantum of energy , while ˆ 𝑎 † creates one quantum of energy . Moreo ver , the quantization procedure is more con v enient using these operators [ 8 – 10 ]. In terms of these operators, w e can define the nor mal ordering and antinor mal order ing. With the f or mer , all ˆ 𝑎 † ’ s precede all ˆ 𝑎 ; with the latter , w e hav e the opposite [ 7 ]. The normal order ing is par ticularl y useful since any e xpression in this ordering is phy sically measurable (for e xample, see Section 3.7 of Ref. [ 11 ]). Based on this understanding, w e are motivated to re wr ite the q uantum expected dynamics in terms of ˆ 𝑎 and ˆ 𝑎 † , then ev aluate the nor mal-ordered equiv alent of the result. C. Main result Instead of summing o v er all possible order ings 𝜔 , let us sum ov er all possible “f orced order ings ” 𝜎 of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . By f orced ordering, we mean sw apping tw o identical operators and considering it as a separate order ing. For e xample, the f orced order ing of ˆ 𝑞 2 ˆ 𝑝 giv es us ˆ 𝑞 ˆ 𝑞 ˆ 𝑝 , ˆ 𝑞 ˆ 𝑞 ˆ 𝑝 , ˆ 𝑞 ˆ 𝑝 ˆ 𝑞 , ˆ 𝑞 ˆ 𝑝 ˆ 𝑞 , ˆ 𝑝 ˆ 𝑞 ˆ 𝑞 , ˆ 𝑝 ˆ 𝑞 ˆ 𝑞 . As shown belo w , the moments appear ing in the final result depend on ( 𝑗 + 𝑘 ) , not their individual v alues. Forced order ing is thus more con v enient, as the number of order ings does not depend on the individual values of 𝑗 and 𝑘 . For a giv en ( 𝑗 , 𝑘 ) , we consider the a verag e ov er the forced orderings 𝜎 of the operators [ 12 ], ˆ 𝑆 𝑗 𝑘 = 1 ( 𝑗 + 𝑘 ) !  𝜎  ˆ 𝑞 𝑗 ˆ 𝑝 𝑘  𝜎 = 𝑖 𝑘 2 ( 𝑗 + 𝑘 ) / 2 ( 𝑗 + 𝑘 ) !  𝜎 𝑗 + 𝑘 Ö 𝑟 = 1  ˆ 𝑎 † + 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 ) ˆ 𝑎  , (6) where 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 ) is the sign of the 𝑟 th factor for the f orced order ing 𝜎 of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 . If the cor responding operator is ˆ 𝑞 , then 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 ) = 1; otherwise, it equals − 1. Ev aluating the product, we obtain the g eneral form ˆ 𝑆 𝑗 𝑘 = 𝑖 𝑘 2 ( 𝑗 + 𝑘 ) / 2 ( 𝑗 + 𝑘 ) !  𝜎 ( 𝑗 + 𝑘 ) / 2  𝑢 = 0 𝑗 + 𝑘 − 2 𝑢  𝑣 = 0 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 ˆ 𝑎 † ( 𝑗 + 𝑘 − 2 𝑢 − 𝑣 ) ˆ 𝑎 𝑣 . (7) Here 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 is a polynomial in 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) , 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 2 ) , . . . , 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑢 + 𝑣 ) , where 𝑟 1 , 𝑟 2 , . . . 𝑟 𝑢 + 𝑣 denote some ( 𝑢 + 𝑣 ) of the ( 𝑗 + 𝑘 ) signs. For e xample, with 𝑗 + 𝑘 = 3, w e hav e 𝜂 𝑗 𝑘 11 𝜎 = 2 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 2 ) + 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 3 ) . T o ar rive at Eq. ( 7 ), we hav e utilized the commutation relation  ˆ 𝑎 , ˆ 𝑎 †  = 1. W e find looking f or the g eneral f orm of 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 difficult, and instead consider the sum ov er 𝜎 for a given ( 𝑢, 𝑣 ) . W e ev aluate the sum with mathematical induction and find that the sum ov er 𝜎 may be wr itten as a product of three quantities:  𝜎 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 = 𝜆 𝑗 𝑘 𝑢𝑣 𝜉 𝑗 𝑘 𝑢𝑣 𝜁 𝑗 𝑘 𝑢𝑣 . (8) 3 When we write out the sum, the term 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 2 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑢 + 𝑣 ) f or some 𝜎 appears exactl y 𝜆 𝑗 𝑘 𝑢𝑣 times in the place of each term of 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 . In general, 𝜆 𝑗 𝑘 𝑢𝑣 = ( 𝑗 + 𝑘 − 𝑢 − 𝑣 ) ! ( 𝑢 + 𝑣 ) ! . (9) Summing like ter ms and factoring out 𝜆 𝑗 𝑘 𝑢𝑣 , we hav e the sum 𝜉 𝑗 𝑘 𝑢𝑣 of all coefficients in 𝜂 𝑗 𝑘 𝑢𝑣 𝜎 , generall y given by the concatenated Bessel-scaled Pascal triangles [ 13 ] 𝜉 𝑗 𝑘 𝑢𝑣 = ( 𝑗 + 𝑘 ) ! 2 𝑢 𝑢 ! 𝑣 ! ( 𝑗 + 𝑘 − 2 𝑢 − 𝑣 ) ! . (10) The quantity 𝜆 𝑗 𝑘 𝑢𝑣 𝜉 𝑗 𝑘 𝑢𝑣 is the total coefficient of each distinct ter m. Summing ov er the distinct ter ms and factoring out the common 𝜆 𝑗 𝑘 𝑢𝑣 𝜉 𝑗 𝑘 𝑢𝑣 , w e hav e 𝜁 𝑗 𝑘 𝑢𝑣 = 𝑗 + 𝑘  𝑟 1 = 1 𝑗 + 𝑘  𝑟 2 = 𝑟 1 + 1 · · · 𝑗 + 𝑘  𝑟 𝑢 + 𝑣 = 𝑟 𝑢 + 𝑣 − 1 + 1 𝑠 ( 𝑗 𝑘 ) 𝑟 1 𝑠 ( 𝑗 𝑘 ) 𝑟 2 . . . 𝑠 ( 𝑗 𝑘 ) 𝑟 𝑢 + 𝑣 ; (11a) 𝜁 𝑗 𝑘 00 = 1 , (11b) where 𝑠 ( 𝑗 𝑘 ) 𝑟 𝑚 can be of any 𝜎 since w e ma y star t with any 𝜎 and sum through all the permutations. W e find it con venient to c hoose 𝜎 such that 𝑆 ( 𝑗 𝑘 ) 𝑟 𝑚 = 1 f or 𝑟 𝑚 = 1 , 2 , . . . , 𝑗 and − 1 other wise. W e obtain 𝜁 𝑗 𝑘 𝑢𝑣 = 𝑢 + 𝑣  𝑚 = 0 ( − 1 ) 𝑚  𝑗 𝑢 + 𝑣 − 𝑚   𝑘 𝑚  , (12) where  𝑎 𝑏  is the binomial coefficient w e tak e to be zero when 𝑎 < 𝑏 or 𝑏 < 0. This summation is kno wn to be the alternating-sign V ander monde con v olution, whose alternative e xpression is [ 14 ] 𝜁 𝑗 𝑘 𝑢𝑣 =  𝑥 𝑢 + 𝑣  ( 1 + 𝑥 ) 𝑗 ( 1 − 𝑥 ) 𝑘 , (13) i.e., the coefficient of 𝑥 𝑢 + 𝑣 in the polynomial ( 1 + 𝑥 ) 𝑗 ( 1 − 𝑥 ) 𝑘 . Substituting Eqs. ( 8 )–( 13 ) into Eq. ( 7 ), w e finally obtain ˆ 𝑆 𝑗 𝑘 = ( 𝑗 + 𝑘 ) / 2  𝑢 = 0 𝑗 + 𝑘 − 2 𝑢  𝑣 = 0 ℎ 𝑗 𝑘 𝑢𝑣 ˆ 𝑎 † ( 𝑗 + 𝑘 − 2 𝑢 − 𝑣 ) ˆ 𝑎 𝑣 ; (14a) ℎ 𝑗 𝑘 𝑢𝑣 = 𝑖 𝑘 2 ( 𝑗 + 𝑘 ) / 2 𝑢 ! 2 𝑢  [ 𝑗 + 𝑘 ] − [ 𝑢 + 𝑣 ] 𝑢   𝑢 + 𝑣 𝑢    𝑥 𝑢 + 𝑣  ( 1 + 𝑥 ) 𝑗 ( 1 − 𝑥 ) 𝑘  . (14b) W e restate here that ˆ 𝑆 𝑗 𝑘 is the W e y l order ing of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 , and the abo v e e xpression giv es its normal-ordered equivalent. D. Symmetry The coefficients ℎ 𝑗 𝑘 𝑢𝑣 satisfy ℎ 𝑗 𝑘 𝑢𝑣 = ( − 1 ) 𝑘 ℎ 𝑗 𝑘 𝑢 ( 𝑗 + 𝑘 − 2 𝑢 − 𝑣 ) ; (15a) ℎ 𝑗 𝑘 𝑢 ( [ 𝑗 + 𝑘 − 2 𝑢 ]/ 2 ) = 0 if 𝑗 and 𝑘 are odd. (15b) One can show that the product of the factors e x cluding [ 𝑥 𝑢 + 𝑣 ] ( 1 + 𝑥 ) 𝑗 ( 1 − 𝑥 ) 𝑘 is identical for both ℎ coefficients in both equations, whence the proper ties are consequences of ho w the polynomial ( 1 + 𝑥 ) 𝑗 ( 1 − 𝑥 ) 𝑘 beha v es under 𝑥 ↦→ − 𝑥 . One ma y argue that these properties are rather trivial, since they translate to the req uirement that ˆ 𝑎 † ( 𝑗 + 𝑘 − 2 𝑢 − 𝑣 ) ˆ 𝑎 𝑣 and ˆ 𝑎 † 𝑣 ˆ 𝑎 𝑗 + 𝑘 − 2 𝑢 − 𝑣 f or m a conjugate transpose pair; accounting for the factors 𝑖 𝑘 and ( − 1 ) 𝑘 , we can see that the requirement arises because we are quantizing a real-valued e xpression. 4 III. CON CLUDIN G REMARKS W e hav e ev aluated the nor mal-ordered equivalent of the W e y l order ing of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 using a brute f orce approach (see also the Supplementary Materials). In the follo wing, w e note sev eral other methods to tackle this problem, which should reproduce our end result or giv e an alternative e xpression. Cahill and Glauber [ 8 ] f or mulated an e xplicit f or mula to wr ite an unev aluated W e y l order ing of ˆ 𝑎 𝑚 ˆ 𝑎 † 𝑛 in ter ms of nor mal ordered e xpressions. Let  ˆ 𝑎 𝑚 ˆ 𝑎 † 𝑛  W be the W e y l order ing of ˆ 𝑎 𝑚 ˆ 𝑎 † 𝑛 (e.g.,  ˆ 𝑎 ˆ 𝑎 †  W =  ˆ 𝑎 ˆ 𝑎 † + ˆ 𝑎 † ˆ 𝑎  / 2). Then,  ˆ 𝑎 𝑚 ˆ 𝑎 † 𝑛  W = min ( 𝑚, 𝑛 )  𝑙 = 0 𝑙 ! 2 𝑙  𝑚 𝑙   𝑛 𝑙  ˆ 𝑎 † ( 𝑛 − 𝑙 ) ˆ 𝑎 ( 𝑚 − 𝑙 ) . (16) One can thus first write the W e yl ordering of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 in terms of  ˆ 𝑎 𝑚 ˆ 𝑎 † 𝑛  W bef ore using the abov e equation. Further more, Blasiak [ 15 ] f ormulated an explicit f ormula f or the nor mal-ordered equiv alent of an y s tring built out of ˆ 𝑎 and ˆ 𝑎 † , called a “boson string”. Let ˆ 𝑋 = ˆ 𝑎 † 𝑟 𝑀 ˆ 𝑎 𝑠 𝑀 ˆ 𝑎 † 𝑟 𝑀 − 1 ˆ 𝑎 𝑠 𝑀 − 1 . . . ˆ 𝑎 † 𝑟 2 ˆ 𝑎 𝑠 2 ˆ 𝑎 † 𝑟 1 ˆ 𝑎 𝑠 1 (17) be a boson string. Then, its nor mal-ordered equiv alent is given b y ˆ 𝑋 =                𝑠 1 + 𝑠 2 + · · · + 𝑠 𝑀  𝑘 = 𝑠 1 𝑆 𝒓 , 𝒔 ( 𝑘 ) ˆ 𝑎 † ( 𝑑 𝑀 + 𝑘 ) ˆ 𝑎 𝑘 , 𝑑 𝑀 ≥ 0; 𝑟 1 + 𝑟 2 + · · · + 𝑟 𝑀  𝑘 = 𝑟 𝑀 𝑆 𝒔 , 𝒓 ( 𝑘 ) ˆ 𝑎 † 𝑘 ˆ 𝑎 ( − 𝑑 𝑀 + 𝑘 ) , 𝑑 𝑀 < 0 , (18) where 𝒓 = ( 𝑟 1 , 𝑟 2 , . . . , 𝑟 𝑀 ) , 𝒔 = ( 𝑠 1 , 𝑠 2 , . . . , 𝑠 𝑀 ) , 𝒓 = ( 𝑟 𝑀 , 𝑟 𝑀 − 1 , . . . , 𝑟 1 ) , 𝒔 = ( 𝑠 𝑀 , 𝑠 𝑀 − 1 , . . . , 𝑠 1 ) , 𝑑 𝑙 = 𝑙  𝑚 = 1 ( 𝑟 𝑚 − 𝑠 𝑚 ) , (19) and 𝑆 𝒓 , 𝒔 ( 𝑘 ) = 1 𝑘 ! 𝑘  𝑗 = 0  𝑘 𝑗  ( − 1 ) 𝑘 − 𝑗 𝑀 Ö 𝑚 = 1 ( 𝑑 𝑚 − 1 + 𝑗 ) 𝑠 𝑚 , (20) where ( 𝑚 ) 𝑛 = 𝑚 ! / ( 𝑚 − 𝑛 ) ! is the falling f actorial. One may rewrite the W e y l order ing of ˆ 𝑞 𝑗 ˆ 𝑝 𝑘 in ter ms of ˆ 𝑎 and ˆ 𝑎 † , and use Blasiak’ s formula abo v e to ev aluate its nor mal-ordered equiv alent. [1] T . Curtr ight, D. Fair lie, and C. Zachos, A Concise T reatise On Quantum Mec hanics In Phase Space (W orld Scientific Publishing Company , 2013). [2] L. Ben Arosh, M. C. Cross, and R. 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[10] H. M. Lim, Algebraic mac hinery of quantization (2025), arXiv:2509.17106 [quant-ph] . [11] C. Gerr y and P . Knight, Introductory Quantum Optics (Cambr idg e Univ ersity Press, 2005). [12] If we assume not to know what values 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 ) ma y take, then we can think of the sum ov er 𝜎 as the sum ov er all per mutations of n 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 ) o —hence our choice of notation. [13] P . Lusc hn y , Concatenated Bessel-scaled Pascal tr iangles (2024), Entry A344911 in The On-Line Ency clopedia of Integer Seq uences. [14] anon (11763/anon), Alternating sign vandermonde conv olution , Mathematics Stack Ex change (2011). [15] P . Blasiak, Combinatorics of boson normal order ing and some applications (2005), arXiv :quant-ph/0507206 [quant-ph] . 5 SUPPLEMENT ARY MA TERIALS A. Obtaining the general form of 𝜆 𝑘 𝑙 𝑝 𝑞 and 𝜉 𝑗 𝑘 𝑝𝑞 As an e xample, let us consider the expansion f or 𝑗 + 𝑘 = 3 f or a giv en per mutation 𝜎 (dropping their corresponding notations in the equation): 3 Ö 𝑟 = 1  ˆ 𝑎 † + 𝑠 𝑟 ˆ 𝑎  = ˆ 𝑎 † 3 + ( 𝑠 1 + 𝑠 2 + 𝑠 3 ) ˆ 𝑎 † 2 ˆ 𝑎 + ( 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 + 𝑠 2 𝑠 3 ) ˆ 𝑎 † ˆ 𝑎 2 + 𝑠 1 𝑠 2 𝑠 3 ˆ 𝑎 3 + ( 2 𝑠 1 + 𝑠 2 ) ˆ 𝑎 † + ( 2 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 ) ˆ 𝑎 (21) W e can immediately see the pattern where all the moments are of orders 𝑗 + 𝑘 (first ro w), then 𝑗 + 𝑘 − 2 (second ro w), 𝑗 + 𝑘 − 4 (does not apply here), . . . , 0 if 𝑗 + 𝑘 is e v en and 1 if odd. The ter ms hav e been grouped b y 𝑝 and 𝑞 . W e hav e ( 𝑝 , 𝑞 ) = ( 0 , 0 ) , ( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) , ( 1 , 0 ) , ( 1 , 1 ) , in that order . W e are interested in the sum o ver all permutations 𝜎 f or the giv en 𝑗 , 𝑘 , 𝑝 , 𝑞 . As a less obvious e xample, let us pick the ter m with ˆ 𝑎 . Let the expression sho wn, ( 2 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 ) ˆ 𝑎 , be the ( 1 , 2 , 3 ) permutation. Then, w e ha v e fiv e other per mutations to write out. The sum is giv en b y (e xplicitly wr iting 0 𝑠 𝜎 ( 2 ) 𝑠 𝜎 ( 3 ) so w e ha ve all the possible ( 𝑝 + 𝑞 ) out of ( 𝑗 + 𝑘 ) signs)  𝜎 2 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 2 ) + 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 3 ) + 0 𝑠 𝜎 ( 2 ) 𝑠 𝜎 ( 3 ) = ( 2 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 + 0 𝑠 2 𝑠 3 ) + ( 2 𝑠 1 𝑠 2 + 𝑠 1 𝑠 2 + 0 𝑠 3 𝑠 2 ) + ( 2 𝑠 2 𝑠 1 + 𝑠 2 𝑠 3 + 0 𝑠 1 𝑠 3 ) + ( 2 𝑠 2 𝑠 3 + 𝑠 2 𝑠 1 + 0 𝑠 3 𝑠 1 ) + ( 2 𝑠 3 𝑠 1 + 𝑠 3 𝑠 2 + 0 𝑠 1 𝑠 2 ) + ( 2 𝑠 3 𝑠 2 + 𝑠 3 𝑠 1 + 0 𝑠 2 𝑠 1 ) (22) Notice how 𝑠 1 𝑠 2 = 𝑠 2 𝑠 1 appears twice in ev er y “slot”. In the first row , it appears in the first slot; in the second to the third row and first column, it appears in the second slot; in the second to the third ro w and second column, it appears in the third slot. It is similar f or 𝑠 1 𝑠 3 and 𝑠 2 𝑠 3 . As such, 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑝 + 𝑞 ) appears a constant number of times in each slot of the 𝜂 𝑗 𝑘 𝑝 𝑞 𝜎 . Let this number be 𝜆 𝑗 𝑘 𝑝 𝑞 . If we no w sum like terms, we g et  𝜎 2 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 2 ) + 𝑠 𝜎 ( 1 ) 𝑠 𝜎 ( 3 ) + 0 𝑠 𝜎 ( 2 ) 𝑠 𝜎 ( 3 ) = [ ( 2 ) ( 2 ) + ( 2 ) ( 1 ) + ( 2 ) ( 0 ) ] 𝑠 1 𝑠 2 + [ ( 2 ) ( 2 ) + ( 2 ) ( 1 ) + ( 2 ) ( 0 ) ] 𝑠 1 𝑠 3 + [ ( 2 ) ( 2 ) + ( 2 ) ( 1 ) + ( 2 ) ( 0 ) ] 𝑠 2 𝑠 3 = 2 ( 2 + 1 + 0 ) ( 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 + 𝑠 2 𝑠 3 ) (23) Each of 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑝 + 𝑞 ) appears twice in the firs t slot with “weight ” 2, twice in the second slot with w eight 1, and twice in the third slot with w eight 0. If we f actor out the common 𝜆 𝑗 𝑘 𝑝 𝑞 = 2, w e hav e a sum ov er the weights, which are just the coefficients of 𝜂 𝑗 𝑘 𝑝 𝑞 𝜎 . This is shared b y ev er y possible 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑝 + 𝑞 ) , so w e can factor out the common sum of weights, which w e denote 𝜉 𝑗 𝑘 𝑝 𝑞 , lea ving out a sum o v er all the 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑝 + 𝑞 ) , which w e denote 𝜁 𝑗 𝑘 𝑝 𝑞 . W e find that 𝜆 𝑗 𝑘 𝑝 𝑞 = ( 𝑗 + 𝑘 ) ! 𝐶 ( 𝑗 + 𝑘 , 𝑝 + 𝑞 ) (24) where ( 𝑗 + 𝑘 ) ! is the number of possible wa ys we can rear rang e 𝑠 1 , 𝑠 2 , . . . , 𝑠 𝑗 + 𝑘 and 𝐶 ( 𝑗 + 𝑘 , 𝑝 + 𝑞 ) is the possible product- of- ( 𝑝 + 𝑞 ) -out-of- ( 𝑗 + 𝑘 ) -signs (i.e. the number of possible terms in the polynomial 𝜂 𝑗 𝑘 𝑝 𝑞 𝜎 ) there is. Mean while, plugging the sum-of-coefficients patter n f or different ( 𝑝 , 𝑞 ) into OEIS, we find that it is giv en by the concatenated Bessel-scaled Pascal triangle [ 13 ], 𝜉 𝑗 𝑘 𝑝 𝑞 = ( 𝑗 + 𝑘 ) ! 2 𝑝 𝑝 ! 𝑞 ! ( 𝑗 + 𝑘 − 2 𝑝 − 𝑞 ) ! (25) 6 B. Evaluating 𝜁 𝑗 𝑘 𝑝𝑞 W e to calculate the sum o v er all 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 1 ) . . . 𝑠 ( 𝑗 𝑘 ) 𝜎 ( 𝑟 𝑝 + 𝑞 ) . This is a sum ov er products of ( 𝑝 + 𝑞 ) signs. The order does not matter , as w e can start with an y order and write out all the per mutations anyw a y . Generalizing from the sum 𝑠 1 𝑠 2 + 𝑠 1 𝑠 3 + 𝑠 2 𝑠 3 in our e xample, w e ha v e the general f orm 𝜁 𝑗 𝑘 𝑝 𝑞 = 𝑗 + 𝑘  𝑟 1 𝑗 + 𝑘  𝑟 2 = 𝑟 1 + 1 · · · 𝑗 + 𝑘  𝑟 𝑝 + 𝑞 = 𝑟 𝑝 + 𝑞 − 1 + 1 𝑠 𝑟 1 𝑠 𝑟 2 . . . 𝑠 𝑟 𝑝 + 𝑞 (26) Since the order of signs does not matter , let us choose 𝑠 𝑝 = ( 1 , 𝑝 = 1 , 2 , . . . , 𝑗 − 1 , 𝑝 = 𝑗 + 1 , 𝑗 + 2 , . . . , 𝑗 + 𝑘 (27) Our strategy is to break up the summation into inter v als where each 𝑠 𝑝 takes the same value. Doing this, ho we ver , requires that the terms exis t in the first place. Here’ s a helpful function: 𝑔 ( 𝑎 , 𝑏 ) = ( 1 , if 𝑎 ≥ 𝑏 0 , other wise (28) As seen below , the ter ms resulting from the par tition make it all the wa y to the end, ev en though it must be zero f or the giv en ( 𝑗 , 𝑘 ) and 𝑝 + 𝑞 . For e xample, with ( 𝑗 , 𝑘 ) = ( 2 , 1 ) , the third summation in “Case 𝑝 + 𝑞 = 2” should not contr ibute to the sum, y et it is there at the end. Using the function 𝑔 , we k eep track of which term ma y contr ibute to the sum by ensur ing that there is at least one ter m in the sum partition—only then is the contr ibution v alid. Let the “completel y par titioned sum” be the ser ies of sums where all 𝑠 𝑝 has taken their v alues. This is where we can mos t easil y “modify” the sum by putting in 𝑔 ( 𝑎 , 𝑏 ) . Case 𝑝 + 𝑞 = 0 Here w e hav e unity . Case 𝑝 + 𝑞 = 1 𝑗 + 𝑘  𝑃 = 1 𝑠 𝑃 = 𝑔 ( 𝑗 , 1 ) 𝑗  𝑃 = 1 ( 1 ) + 𝑔 ( 𝑘 , 1 ) 𝑗 + 𝑘  𝑃 = 𝑗 + 1 ( − 1 ) = 𝑔 ( 𝑗 , 1 ) 𝑗 − 𝑔 ( 𝑘 , 1 ) 𝑘 (29) Case 𝑝 + 𝑞 = 2 𝑗 + 𝑘  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑠 𝑃 𝑠 𝑄 = 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 ( 1 ) 𝑠 𝑄 + 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 ( − 1 ) 𝑠 𝑄 = 𝑔 ( 𝑗 , 2 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 ( 1 ) ( 1 ) + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 1 ) 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑗 + 1 ( 1 ) ( − 1 ) + 𝑔 ( 𝑘 , 2 ) 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 ( − 1 ) ( − 1 ) = 𝑔 ( 𝑗 , 2 ) 𝑗 ( 𝑗 − 1 ) 2 − 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 1 ) 𝑗 𝑘 + 𝑔 ( 𝑘 , 2 ) 𝑘 ( 𝑘 − 1 ) 2 (30) 7 Case 𝑝 + 𝑞 = 3 𝑗 + 𝑘  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑠 𝑃 𝑠 𝑄 𝑠 𝑅 = 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( 1 ) 𝑠 𝑄 𝑠 𝑅 + 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( − 1 ) 𝑠 𝑄 𝑠 𝑅 = 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( 1 ) ( 1 ) 𝑠 𝑅 + 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑗 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( 1 ) ( − 1 ) 𝑠 𝑅 + 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( − 1 ) ( − 1 ) 𝑠 𝑅 = 𝑔 ( 𝑗 , 3 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 ( 1 ) ( 1 ) ( 1 ) + 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 1 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑗 + 1 ( 1 ) ( 1 ) (− 1 ) + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 2 ) 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑗 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( 1 ) ( − 1 ) (− 1 ) + 𝑔 ( 𝑘 , 3 ) 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 ( − 1 ) ( − 1 ) (− 1 ) = 𝑔 ( 𝑗 , 3 ) 𝑗 ( 𝑗 − 1 ) ( 𝑗 − 2 ) 6 − 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 1 ) 𝑗 ( 𝑗 − 1 ) 𝑘 2 + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 2 ) 𝑗 𝑘 ( 𝑘 − 1 ) 2 − 𝑔 ( 𝑘 , 3 ) 𝑘 ( 𝑘 − 1 ) ( 𝑘 − 2 ) 6 (31) Case 𝑝 + 𝑞 = 4 𝑗 + 𝑘  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 𝑠 𝑃 𝑠 𝑄 𝑠 𝑅 𝑠 𝑇 = 𝑔 ( 𝑗 , 4 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 𝑗  𝑇 = 𝑅 + 1 ( 1 ) ( 1 ) ( 1 ) ( 1 ) + 𝑔 ( 𝑗 , 3 ) 𝑔 ( 𝑘 , 1 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑗 + 1 ( 1 ) ( 1 ) ( 1 ) ( − 1 ) + 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 2 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑗 + 1 𝑗 + 𝑘  𝑇 = 𝑟 + 1 ( 1 ) ( 1 ) (− 1 ) ( − 1 ) + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 3 ) 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑗 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑟 + 1 ( 1 ) ( − 1 ) (− 1 ) ( − 1 ) + 𝑔 ( 𝑘 , 4 ) 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 ( − 1 ) ( − 1 ) (− 1 ) ( − 1 ) = 𝑔 ( 𝑗 , 4 ) 𝑗 ( 𝑗 − 1 ) ( 𝑗 − 2 ) ( 𝑗 − 3 ) 24 − 𝑔 ( 𝑗 , 3 ) 𝑔 ( 𝑘 , 1 ) 𝑗 ( 𝑗 − 1 ) ( 𝑗 − 2 ) 𝑘 6 + 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 2 ) 𝑗 ( 𝑗 − 1 ) 𝑘 ( 𝑘 − 1 ) 4 − 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 3 ) 𝑗 𝑘 ( 𝑘 − 1 ) ( 𝑘 − 2 ) 6 + 𝑔 ( 𝑘 , 4 ) 𝑘 ( 𝑘 − 1 ) ( 𝑘 − 2 ) ( 𝑘 − 3 ) 24 (32) 8 Case 𝑝 + 𝑞 = 5 𝑗 + 𝑘  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 𝑗 + 𝑘  𝑈 = 𝑇 + 1 𝑠 𝑃 𝑠 𝑄 𝑠 𝑅 𝑠 𝑇 𝑠 𝑈 = 𝑔 ( 𝑗 , 5 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 𝑗  𝑇 = 𝑅 + 1 𝑗  𝑈 = 𝑇 + 1 ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( 1 ) + 𝑔 ( 𝑗 , 4 ) 𝑔 ( 𝑘 , 1 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 𝑗  𝑇 = 𝑅 + 1 𝑗 + 𝑘  𝑈 = 𝑗 + 1 ( 1 ) ( 1 ) ( 1 ) ( 1 ) ( − 1 ) + 𝑔 ( 𝑗 , 3 ) 𝑔 ( 𝑘 , 2 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑗 + 1 𝑗 + 𝑘  𝑈 = 𝑇 + 1 ( 1 ) ( 1 ) ( 1 ) ( − 1 ) ( − 1 ) + 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 3 ) 𝑗  𝑃 = 1 𝑗  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑗 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 𝑗 + 𝑘  𝑈 = 𝑇 + 1 ( 1 ) ( 1 ) (− 1 ) ( − 1 ) ( − 1 ) + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 4 ) 𝑗  𝑃 = 1 𝑗 + 𝑘  𝑄 = 𝑗 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 𝑗 + 𝑘  𝑈 = 𝑇 + 1 ( 1 ) ( − 1 ) (− 1 ) ( − 1 ) ( − 1 ) + 𝑔 ( 𝑘 , 5 ) 𝑗 + 𝑘  𝑃 = 𝑗 + 1 𝑗 + 𝑘  𝑄 = 𝑃 + 1 𝑗 + 𝑘  𝑅 = 𝑄 + 1 𝑗 + 𝑘  𝑇 = 𝑅 + 1 𝑗 + 𝑘  𝑈 = 𝑇 + 1 ( − 1 ) ( − 1 ) (− 1 ) ( − 1 ) ( − 1 ) = 𝑔 ( 𝑗 , 5 ) 𝑔 ( 𝑘 , 0 ) 𝑗 ! 𝑘 ! 5! ( 𝑗 − 5 ) !0! ( 𝑘 − 0 ) ! − 𝑔 ( 𝑗 , 4 ) 𝑔 ( 𝑘 , 1 ) 𝑗 ! 𝑘 ! 4! ( 𝑗 − 4 ) !1! ( 𝑘 − 1 ) ! + 𝑔 ( 𝑗 , 3 ) 𝑔 ( 𝑘 , 2 ) 𝑗 ! 𝑘 ! 3! ( 𝑗 − 3 ) !2! ( 𝑘 − 2 ) ! − 𝑔 ( 𝑗 , 2 ) 𝑔 ( 𝑘 , 3 ) 𝑗 ! 𝑘 ! 2! ( 𝑗 − 2 ) !3! ( 𝑘 − 3 ) ! + 𝑔 ( 𝑗 , 1 ) 𝑔 ( 𝑘 , 4 ) 𝑗 ! 𝑘 ! 1! ( 𝑗 − 1 ) !4! ( 𝑘 − 4 ) ! − 𝑔 ( 𝑗 , 0 ) 𝑔 ( 𝑘 , 5 ) 𝑗 ! 𝑘 ! 0! ( 𝑗 − 0 ) !5! ( 𝑘 − 5 ) ! (33) General formula 𝜁 𝑗 𝑘 𝑝 𝑞 = 𝑗 + 𝑘  𝑟 1 = 1 𝑗 + 𝑘  𝑟 2 = 𝑟 1 + 1 · · · 𝑗 + 𝑘  𝑟 𝑝 + 𝑞 = 𝑟 𝑝 + 𝑞 − 1 + 1 𝑠 𝑟 1 𝑠 𝑟 2 . . . 𝑠 𝑟 𝑝 + 𝑞 = 𝑝 + 𝑞  𝑚 = 0 ( − 1 ) 𝑚 𝑔 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) 𝑗 ! ( 𝑝 + 𝑞 − 𝑚 ) ! ( 𝑗 − 𝑝 − 𝑞 + 𝑚 ) ! 𝑔 ( 𝑘 , 𝑚 ) 𝑘 ! 𝑚 ! ( 𝑘 − 𝑚 ) ! = 𝑝 + 𝑞  𝑚 = 0 ( − 1 ) 𝑚 𝑔 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) 𝐶 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) 𝑔 ( 𝑘 , 𝑚 ) 𝐶 ( 𝑘 , 𝑚 ) (34) The 𝑚 th term is zero whenev er the difference betw een the firs t and second argument in the binomial coefficient is less than zero, implying that w e can simplify this f or mula as 𝜁 𝑗 𝑘 𝑝 𝑞 = 𝑝 + 𝑞  𝑚 = 0 ( − 1 ) 𝑚 𝛾 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) 𝛾 ( 𝑘 , 𝑚 ) (35) where 𝛾 ( 𝑎 , 𝑏 ) = 𝑔 ( 𝑎 , 𝑏 ) 𝐶 ( 𝑎 , 𝑏 ) = ( 𝐶 ( 𝑎 , 𝑏 ) , if 𝑎 ≥ 𝑏 0 , otherwise (36) W e can also write the result in ter ms of nonzero contributions only , though the summation limit is not as pretty : 𝜁 𝑗 𝑘 𝑝 𝑞 = min ( 𝑘 , 𝑝 + 𝑞 )  𝑚 = max ( 0 , 𝑝 + 𝑞 − 𝑗 ) ( − 1 ) 𝑚 𝐶 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) 𝐶 ( 𝑘 , 𝑚 ) (37) where 𝑝 + 𝑞 − 𝑗 gives the lo wes t 𝑚 f or whic h 𝛾 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) = 𝐶 ( 𝑗 , 𝑝 + 𝑞 − 𝑚 ) (the second argument decreases as 𝑚 increases) and 𝑘 gives the highes t 𝑚 f or which 𝛾 ( 𝑘 , 𝑚 ) = 𝐶 ( 𝑘 , 𝑚 ) (the second argument increases with 𝑚 ).