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In mathematics, Dixon's identity (or Dixon's theorem or Dixon's formula) is any of several different but closely related identities proved by A. C. Dixon, some involving finite sums of products of three binomial coefficients, and some evaluating a hypergeometric sum. These identities famously follow from the MacMahon Master theorem, and can now be routinely proved by computer algorithms Template:Harv.

Statements

The original identity, from Template:Harv, is

\sum_{k = - a}^{a} (- 1)^{k} {(\binom{2 a}{k + a})}^{3} = \frac{(3 a)!}{(a!)^{3}} .

A generalization, also sometimes called Dixon's identity, is

\sum_{k = - a}^{a} (- 1)^{k} (\binom{a + b}{a + k}) (\binom{b + c}{b + k}) (\binom{c + a}{c + k}) = \frac{(a + b + c)!}{a! b! c!}

where a, b, and c are non-negative integers Template:Harv. The sum on the left can be written as the terminating well-poised hypergeometric series

(\binom{b + c}{b - a}) (\binom{c + a}{c - a})_{3} F_{2} (- 2 a, - a - b, - a - c; 1 + b - a, 1 + c - a; 1)

and the identity follows as a limiting case (as a tends to an integer) of Dixon's theorem evaluating a well-poised ₃F₂ generalized hypergeometric series at 1, from Template:Harv:

_{3} F_{2} (a, b, c; 1 + a - b, 1 + a - c; 1) = \frac{Γ (1 + a / 2) Γ (1 + a / 2 - b - c) Γ (1 + a - b) Γ (1 + a - c)}{Γ (1 + a) Γ (1 + a - b - c) Γ (1 + a / 2 - b) Γ (1 + a / 2 - c)} .

This holds for Re(1 + Template:Fraca − b − c) > 0. As c tends to −∞ it reduces to Kummer's formula for the hypergeometric function ₂F₁ at −1. Dixon's theorem can be deduced from the evaluation of the Selberg integral.

q-analogues

A q-analogue of Dixon's formula for the basic hypergeometric series in terms of the q-Pochhammer symbol is given by

_{4} ϕ_{3} [\begin{matrix} a & - q a^{1 / 2} & b & c \\ - a^{1 / 2} & a q / b & a q / c \end{matrix}; q, q a^{1 / 2} / b c] = \frac{(a q, a q / b c, q a^{1 / 2} / b, q a^{1 / 2} / c; q)_{\infty}}{(a q / b, a q / c, q a^{1 / 2}, q a^{1 / 2} / b c; q)_{\infty}}

where |qa^1/2/bc| < 1.

References

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Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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+In mathematics, '''Dixon's identity''' (or '''Dixon's theorem''' or '''Dixon's formula''') is any of several different but closely related identities proved by [[Alfred Dixon|A. C. Dixon]], some involving finite sums of products of three [[binomial coefficient]]s, and some evaluating a [[hypergeometric sum]].  These identities famously follow from the [[MacMahon Master theorem]], and can now be routinely proved by computer algorithms {{harv|Ekhad|1990}}.
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+==Statements==
+The original identity, from {{harv|Dixon|1891}}, is
+:<math>\sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^3 =\frac{(3a)!}{(a!)^3}.</math>
+A generalization, also sometimes called Dixon's identity, is
+:<math>\sum_{k=-a}^a(-1)^k{a+b\choose a+k} {b+c\choose b+k}{c+a\choose c+k}  = \frac{(a+b+c)!}{a!b!c!}</math>
+where ''a'', ''b'', and ''c'' are non-negative integers {{harv|Wilf|1994|p=156}}.
+The sum on the left can be written as the terminating well-poised hypergeometric series
+:<math>{b+c\choose b-a}{c+a\choose c-a}{}_3F_2(-2a,-a-b,-a-c;1+b-a,1+c-a;1)</math>
+and the identity follows as a limiting case (as ''a'' tends to an integer) of
+Dixon's theorem evaluating  a well-poised <sub>3</sub>''F''<sub>2</sub> [[generalized hypergeometric series]] at 1, from {{harv|Dixon|1902}}:
+:<math>\;_3F_2 (a,b,c;1+a-b,1+a-c;1)=
+\frac{\Gamma(1+a/2)\Gamma(1+a/2-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}
+{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}.</math>
+This holds for Re(1 + {{frac|1|2}}''a'' &minus; ''b'' &minus; ''c'') &gt; 0. As ''c'' tends to &minus;∞ it reduces to [[Kummer's formula]] for the hypergeometric function <sub>2</sub>F<sub>1</sub> at &minus;1. Dixon's theorem can be deduced from the evaluation of the [[Selberg integral]].
+==q-analogues==
+A ''q''-analogue of Dixon's formula for the [[basic hypergeometric series]] in terms of the [[q-Pochhammer symbol]] is given by
+:<math>\;_{4}\phi_3 \left[\begin{matrix}
+a & -qa^{1/2} & b & c \\
+&-a^{1/2} & aq/b & aq/c \end{matrix}
+; q,qa^{1/2}/bc \right] =
+\frac{(aq,aq/bc,qa^{1/2}/b,qa^{1/2}/c;q)_\infty}{(aq/b,aq/c,qa^{1/2},qa^{1/2}/bc;q)_\infty}
+</math>
+where |''qa''<sup>1/2</sup>/''bc''| &lt; 1.
+==References==
+{{reflist}}
+*{{citation |first=A.C. |last=Dixon | authorlink=Alfred Cardew Dixon | title= On the sum of the cubes of the coefficients in a certain expansion by the binomial theorem |journal= [[Messenger of Mathematics]] |volume=20 | pages= 79–80 |year=1891 | jfm=22.0258.01  }}
+*{{citation |first=A.C. |last=Dixon | authorlink=Alfred Cardew Dixon |title= Summation of a certain series |journal= Proc. London Math. Soc. |volume=35 |issue=1 |pages= 284–291 |year=1902 | doi=10.1112/plms/s1-35.1.284 | jfm=34.0490.02  }}
+*{{Citation | last1=Ekhad | first1=Shalosh B. | authorlink=Shalosh Ekhad | title=A very short proof of Dixon's theorem | doi=10.1016/0097-3165(90)90014-N | mr=1051787 | zbl=0707.05007 | year=1990 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=54 | issue=1 | pages=141–142}}
+*{{Citation | last1=Gessel | first1=Ira | last2=Stanton | first2=Dennis | title=Short proofs of Saalschütz's and Dixon's theorems | doi=10.1016/0097-3165(85)90026-3 | mr=773560 | zbl=0559.05008  | year=1985 | journal=Journal of Combinatorial Theory. Series A | issn=1096-0899 | volume=38 | issue=1 | pages=87–90}}
+*{{Citation | last1=Ward | first1=James | title=100 years of Dixon's identity | mr=1185413 | zbl=0795.01009 | year=1991 | journal=Irish Mathematical Society Bulletin | issn=0791-5578 | issue=27 | pages=46–54}}
+* {{citation | last=Wilf | first=Herbert S. | authorlink=Herbert Wilf | title=Generatingfunctionology | edition=2nd | location=Boston, MA | publisher=Academic Press | year=1994 | isbn=0-12-751956-4 | zbl=0831.05001 }}
+[[Category:Enumerative combinatorics]]
+[[Category:Factorial and binomial topics]]
+[[Category:Hypergeometric functions]]
+[[Category:Mathematical identities]]

Antoine equation: Difference between revisions

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