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Even , Odd u .. v  u .: v  

u .. v  ↔  (u + u&v) % 2:
u .: v  ↔  (u - u&v) % 2:
 

In the most commonly used case, v is arithmetic negation, and f=: u .. v is therefore f=: (u + u&-) % 2: ; that is, one-half the sum of u y and u -y . The resulting function is therefore even in the sense that f y ↔ f -y for any y ; its graph is reflected in the vertical axis. Similarly, u .: - is odd (f y ↔ -f-y), and its graph is reflected in the origin. Less commonly, v is matrix transpose (|:), and may be any monadic function.
   y=: _2 _1 0 1 2
   1 2 3 4 5 & p. y            NB. Polynomial with odd and even terms
57 3 1 15 129
 
   1 2 3 4 5 & p. .. - y       NB. Even part of polynomial
93 9 1 9 93
   
   1 0 3 0 5 & p. y            NB. Polynomial with even terms only
93 9 1 9 93

   E=: .. -                    NB. Even adverb
   O=: .: -                    NB. Odd adverb
   d=: 5j2&":@,.&.>            NB. Display as columns with two digits

   d (5&o. ; ^O ; 6&o. ; ^E ; ^ ; (^E + ^O) ; 2&o. ; ^@j.E) y
+-----+-----+-----+-----+-----+-----+-----+-----+
|_3.63|_3.63| 3.76| 3.76| 0.14| 0.14|_0.42|_0.42|
|_1.18|_1.18| 1.54| 1.54| 0.37| 0.37| 0.54| 0.54|
| 0.00| 0.00| 1.00| 1.00| 1.00| 1.00| 1.00| 1.00|
| 1.18| 1.18| 1.54| 1.54| 2.72| 2.72| 0.54| 0.54|
| 3.63| 3.63| 3.76| 3.76| 7.39| 7.39|_0.42|_0.42|
+-----+-----+-----+-----+-----+-----+-----+-----+

   m=: ?. 4 4 $ 9
   (] ; (] .. |:) ; (] .: |:)) m
+-------+-----------+-------------+
|3 8 8 4|  3 5 6 2.5|   0  3 2 1.5|
|2 0 2 7|  5 0 2   4|  _3  0 0   3|
|4 2 3 5|  6 2 3   6|  _2  0 0  _1|
|1 1 7 2|2.5 4 6   2|_1.5 _3 1   0|
+-------+-----------+-------------+

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