I’ve been very quiet here over the last couple of weeks. My mother, Delia Maria Johnson, already in hospital since 5th November or so, took a turn for the worse and began a rapid decline. She died peacefully after some days, and to be honest I’ve really not been myself since then.
There’s an extra element to the sense of loss when (as it approaches) you are powerless to do anything because of being thousands of miles away. On the plus side, because of the ease of using video calls, and with the help of my sister being there, I was able to be somewhat present during what turned out to be the last moments when she was aware of people around her, and therefore was able to tell her I loved her one last time.
Rather than charging across the world on planes, trains, and in automobiles, probably being out of reach during any significant changes in the situation (the doctors said I would likely not make it in time) I did a number of things locally that I am glad I got to do.
It began with visiting (and sending a photo from) the Santa Barbara mission, a place she dearly loved and was unable to visit again after 2019, along with the pier. These are both places we walked together so much back when I first lived here in what feels like another life.
Then, two nights before mum passed away, but well after she’d seemed already beyond reach of anyone, although perhaps (I’d like to think) still able to hear things, my sister contacted me from her bedside asking if I’d like to read mum a psalm, perhaps one of her favourites, 23 or 91. At first I thought she was already planning the funeral, and expressed my surprise at this since mum was still alive and right next to her. But I’d misunderstood, and she’d in fact had a rather great idea. This suggestion turned into several hours of, having sent on recordings of the two psalms, my digging into the poetry shelf in the study and discovering long neglected collections through which I searched (sometimes accompanied by my wife and son) for additional things to read. I recorded some and sent them along, as well as one from my son, I’m delighted to say. Later, the whole thing turned into me singing various songs while playing my guitar and sending recordings of those along too.
Incidentally, the guitar-playing was an interesting turn of events since not many months ago I decided after a long lapse to start playing guitar again, and try to move the standard of my playing (for vocal accompaniment) to a higher level than I’d previously done, by playing and practicing for a little bit on a regular basis. I distinctly recall thinking at one point during one practice that it would be nice to play for mum, although I did not imagine that playing to her while she was on her actual death-bed would be the circumstance under which I’d eventually play for her, having (to my memory) never directly done so back when I used to play guitar in my youth. (Her overhearing me picking out bits of Queen songs behind my room door when I was a teenager doesn’t count as direct playing for her.)
Due to family circumstances I’ll perhaps go into another time, Click to continue reading this post
, and how they generically satisfy a three-term recurrence relation (or recursion relation):
th one is 
where 
is called the “weight function” (it has some suitable properties we won’t worry about here). In the case of random matrix models we have 
for some potential 
(here 
is the size of the matrix; in this problem it is just a normalisation choice – you can just as well absorb it into the potential).
are some positive non-vanishing normalisation constants. Ok now we are ready for the proof.
.
for 
or 
just tells you the definition of the recursion coefficients 
and 
in terms of ratios of inner products for those 
, and for 
and 
.
, which vanishes since the degree of the first entry, 
, is less than 
. Therefore the inner product says 
in all those cases, which means that 
for 
.
. But you can also write this as 
and use the recursion relation 
, and all these terms vanish in the integral except the last, and so we get 
.
up there again.)
